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## About Secant Square X Formula

Secant is one of the functions of Trigonometry which are applied on the right-angled triangle along with sine, cosine, tangent, cosecant, and cotangent and it can be derived by dividing the hypotenuse with the adjacent side of the triangle. Though the secant function is not used much frequently, it is still useful to understand the other Trigonometric functions. In this article, we will learn about the formula of sec square x along with its derivation and some examples.

You will learn how it can be derived and used in Trigonometry and with this how can you solve the advanced examples of trigonometry chapters.

**Introduction **

Trigonometry functions are used on a right-angled triangle and in a right-angled triangle. Secant is considered as the value of an angle which is derived when we divide the length of the hypotenuse with the length of its adjacent side. It is abbreviated as “sec” in formulas of trigonometry and can be derived as a reciprocal of “cosine”. In trigonometry, the most common functions which are used are sine, cosine and tangent whereas the other three are rarely used that are cosecant, secant and cotangent and sometimes, even sometimes most of the calculators do not include buttons for these three functions and also you will find them missing in software functions libraries as well. Secant can be written as follows:

Sec x = H/A

Where H is hypotenuse & A is an adjacent side.

**Sec Square x Formula**

If we talk about the secant, then in trigonometric functions they are usually involved in the form of squares in expressions as well as equations. We can simplify the equations as well expressions only by transferring the functions of the secant square into some equivalent form. Thus, it becomes more important in order to learn about the secant square functions so that other advanced functions of Trigonometry can also be understood. It is mostly used in two cases such as when the secant square function is equal to the summation of one as well as tangent square function and on the other hand, we also can say that the sum of tangent square function and one is also simplified as secant square function.

The formula of secant square function or we can say sec square x formula can be written in the following two forms which are mentioned below. The formula of sec square x in mathematics can be written in any of these two forms:

- Sec
^{2}x = 1 + tan^{2}x - Sec
^{2}A = 1 + tan^{2}A

**Formula Proof**

Let suppose that “theta” is a symbol that represents the angles of a right-angled triangle. In Trigonometry, secant, as well as a tangent, can be written in the form of sec theta and tan theta respectively. According to the Pythagorean identity of secant as well as tangent functions, in trigonometry secant and tangent functions can be written in the following form:

Sec^{2}θ – tan^{2}θ = 1

Therefore, Sec^{2}θ = 1 + tan^{2}θ

Thus, this is how the secant square x formula can be derived and proved. Through this, we can say that the secant square function is equal to the summation of one as well as the tangent square function.

**Examples**

Let’s solve some examples with the help of the secant square x formula which are given below:

**Example 1: **What is the value of sec square x, if tan x = 2/5?

**Solution:** Sec^{2}x = 1 + tan^{2}x

= 1 + (2/5)^{2}

= 1 + 4/25

= (25 + 4) / 25

= 29/25

Hence, Sec^{2}x = 29/25

**Example 2: **What is the value of Sec square x, if tan x = 1/3?

**Solution:** Sec^{2}x = 1 + tan^{2}x

= 1 + (4/5)^{2}

= 1 + 1/9

= (10) / 9

Hence, Sec^{2}x = 10/9

**Example 3: **find the value of sec^{2}x, if the value of cot x = 2/5

**Solution:**

Given, cot x = 2/5

tan x = 1/cot x = 5/2

sec^{2}x = 1 + tan^{2}x

= 1 + (5/2)^{2}

= 1 + 25/4

= 29/4

**Example 4: **Simplify: (1 + tan^{2}A) (1 – sin^{2}A)

**Solution:**

= (1 + tan^{2}A)(1 – sin^{2}A)

= sec^{2}A . cos^{2}A

= (1/cos^{2}A) . cos^{2}A

= 1

**Conclusion**

Thus, to sum the discussion of the sec square x formula, we can say that it can be easily derived from the Pythagorean functions of mathematics and it is equal to the sum of tangent square functions and one. It can be directly used where such expression you can see in the questions or otherwise such expressions can be made in order to reach this form so that formula of sec square x, can easily be used in order to reach the final answer for example in one of the examples, we find out the value of tan x from cot x in order to reach sec^{2}x value.

## FAQs on Secant Square X Formula

Q1. What Do You Mean by Secant in Trigonometry?

Answer. The secant function of an angle is used in a right-angled triangle where it is derived by dividing the hypotenuse and the adjacent side and it is abbreviated as “sec” and written as Sec = H/A, where H means hypotenuse and A means adjacent side.

Q2. What Is the Secant Square X Formula?

Answer. It is equal to the summation of one and tangent square function or we also can say that one plus tangent square function can also be simplified as secant squared function.

Q3. What are the Popular Forms Sec Square X Formula?

Answer. It can be written in two popular forms. First, with the respect of ‘x’ and second with the respect of ‘A’. The formula can be expressed as Sec^{2}x = 1 + tan^{2}x as well as Sec^{2}A = 1 + tan2A.

## Examples of Cos squared theta formula

**Example 1: What is the value of sec square x, if tan x = ⅗?**

Solution: **Sec ^{2} x = 1 + tan^{2} x**

= 1 + (⅗)^{2}

= 1 + 9/25

= (25 + 9) / 25

= 34/25

Sec^{2} x = 34/25

**Example 2: What is the value of Sec square x, if tan x = 4/5?**

Solution: **Sec ^{2} x = 1 + tan^{2} x**

= 1 + (4/5)^{2}

= 1 + 16/25

= (41) / 25

Sec^{2} x = 41/25

**Example 3: If cot x = ⅔, then find the value of sec ^{2}x.**

Solution:

Given,

cot x = ⅔

tan x = 1/cot x = 3/2

sec^{2}x = 1 + tan^{2}x

= 1 + (3/2)^{2}

= 1 + 9/4

= 13/4

**Example 4: Simplify (1 + tan ^{2}A) (1 – sin A) (1 + sin A).**

Solution:

(1 + tan^{2}A) (1 – sin A) (1 + sin A)

= (1 + tan^{2}A)(1 – sin^{2}A)

= sec^{2}A . cos^{2}A

= (1/cos^{2}A) . cos^{2}A

= 1

**Secant Trigonometric Ratio**

The ratio of the lengths of any two sides of a right triangle is called a trigonometric ratio. In trigonometry, these ratios link the ratio of sides of a right triangle to the angle. The secant ratio is expressed as the ratio of the hypotenuse (longest side) to the side corresponding to a given angle in a right triangle. It is the reciprocal of the cosine ratio and is denoted by the abbreviation sec.

If θ is the angle that lies between the base and hypotenuse of a right-angled triangle then,

**sec θ = Hypotenuse/Base = 1/cos θ**

Here, hypotenuse is the longest side of right triangle and base is the side adjacent to the angle.

**Secant Square x Formula**

The secant square x ratio is denoted by the abbreviation sec^{2} x. It’s a trigonometric function that returns the square of the secant function value for an angle x. The period of the function sec x is 2π, but the period of sec^{2} x is π. Its formula is equivalent to the sum of unity and the tangent square function.

**Formula **

sec^{2}x = 1 + tan^{2}xwhere,

x is one of the angles of the right triangle,

tan x is the tangent ratio for angle x.

**Derivation**

The formula for secant square x is derived by using the identity of sum of squares of sine and cosine ratios.

We know, sin^{2} x + cos^{2} x = 1.

Dividing both sides by cos^{2} x, we get

(sin^{2} x/cos^{2} x) + (cos^{2} x/cos^{2} x) = 1/cos^{2} x

tan^{2} x + 1 = sec^{2} x

=> sec^{2} x = 1 + tan^{2} x

This derives the formula for secant square x ratio.

**Sample Problems**

**Problem 1. If tan x = 3/4, find the value of sec ^{2} x using the formula.**

**Solution:**

We have, tan x = 3/4.

Using the formula we get,

sec^{2} x = 1 + tan^{2} x

= 1 + (3/4)^{2}

= 1 + 9/16

= 25/16

**Problem 2. If tan x = 12/5, find the value of sec ^{2} x using the formula.**

**Solution:**

We have, tan x = 12/5.

Using the formula we get,

sec

^{2}x = 1 + tan^{2}x= 1 + (12/5)

^{2}= 1 + 144/25

= 169/25

**Problem 3. If sin x = 8/17, find the value of sec ^{2} x using the formula.**

**Solution:**

We have, sin x = 8/17.

Find the value of cos x using the formula sin^{2} x + cos^{2} x = 1.

cos x = √(1 – (64/289))

= √(225/289)

= 15/17

So, tan x = sin x/cos x = 8/15

Using the formula we get,

sec^{2} x = 1 + tan^{2} x

= 1 + (8/15)^{2}

= 1 + 64/225

= 289/225

**Problem 4. If cot x = 8/15, find the value of sec ^{2} x using the formula.**

**Solution:**

We have, cot x = 8/15.

So, tan x = 1/cot x = 15/8

Using the formula we get,

sec

^{2}x = 1 + tan^{2}x= 1 + (15/8)

^{2}= 1 + 225/64

= 289/64

**Problem 5. If cos x = 12/13, find the value of sec ^{2} x using the formula.**

**Solution:**

We have, cos x = 12/13.

Find the value of sin x using the formula sin^{2} x + cos^{2} x = 1.

sin x = √(1 – (144/169))

= √(25/169)

= 5/13

So, tan x = sin x/cos x = 5/12

Using the formula we get,

sec^{2} x = 1 + tan^{2} x

= 1 + (5/12)^{2}

= 1 + 25/144

= 169/144

## Secant Square x Formula: Proof & Solved Examples

Secant is one of the **Trigonometry** functions that may be used to a right-angled triangle, alongside sine, **cosine**, tangent, cosecant, and cotangent. It can be calculated by dividing the hypotenuse by the triangle’s adjacent side. Even though the secant function is rarely utilised, it is important to learn it for the other **Trigonometric functions**. **Trigonometric identities** are widely used by students and professionals as they are useful for right-angle triangles. In right-angled triangles, you can calculate and find **trigonometric ratios** and identities very easily as the sides and angles are fixed due to angles. Moreover, these identities have a wide application in practical lives as well. Engineers and surveyors use trigonometric identities to calculate the height of a building, find the location of the sailing ship, draw topographical maps on paper, and many more.

**What is Secant?**

Trigonometry functions are utilised on a right-angled triangle and in a right-angled triangle. The value of an angle obtained by dividing the length of the hypotenuse by the length of its neighbouring side is known as secant. It is abbreviated as “sec” in trigonometry calculations and may be calculated as the reciprocal of “cosine.” The most often used functions in trigonometry are sine, cosine, and tangent, although the other three, cosecant, secant, and cotangent, are rarely utilised. Most calculators do not contain buttons for these three functions, and you will also find them absent in software functions libraries. Secant can be written in the following way:

Sec x = H/A

The hypotenuse is H, while the next side is A.

**Trigonometric Ratios**

The core of trigonometry is made up of six ratios. These are the following:

- Cosine (cos)
- Sine (sin)
- Tangent (tan)
- Cotangent (cot)
- Cosecant (cosec)
- Secant (sec)

Three of the six trigonometric ratios are fundamental, whereas the other three are derived. The cosine ratio is used to calculate the secant. In a right-angled triangle, the secant of an angle equals the ratio of the hypotenuse and neighbouring side lengths. It is abbreviated as ‘sec’ in formulae. It is similar to sine and cosine in that it has a period of 2π.

**Secant Square x Formula **

They are commonly used in the form of squares in expressions and equations in trigonometric functions. We reduce the equations and statements by converting the secant square functions into comparable forms. As a result, learning about secant square functions becomes increasingly crucial to comprehend other advanced Trigonometry functions. It’s typically utilised in two situations, one when the secant square function is equal to the sum of one and tangent square function, and second when the sum of tangent square function and one is also simplified as secant square function.

The formula for the secant square function, often known as the sec square x formula, can be stated in two different ways, as shown below. In mathematics, the formula for sec square x can be expressed in one of two ways:

**sec2 A = 1 + tan2 A****sec****2****x = 1 + tan****2****x**

**Proof For Secant Square x Formula**

Let’s pretend that “theta” is a symbol for the angles of a right-angled triangle. A secant and a tangent are represented in trigonometry by the letter’s sec theta and tan theta, respectively. Secant and tangent functions can be represented in the following manner in trigonometry, according to the Pythagorean identity of secant and tangent functions:

sec2 θ – tan2 θ = 1

Therefore, sec2 θ = 1 + tan2 θ

As a result, the secant square x formula may be deduced and proven in this manner. We may state that the secant square function is equal to the summation of one and the tangent square function due to this.

**Other Relations**

Soem other important trigonometric relations are as follows:

- sec(x) = tan(x) / sin(x)
- sec(x) = csc(x) / cot(x)
- sec (x) = 1 / cos (x)
- Sec (– θ) = + Sec θ
- Sec (90 + θ) = Cosec θ
- Sec (90 + θ) = Cosec θ
- Sec (θ + 2πn) = sec θ
- Sec-1 (-x) = -sec-1 x
- Sec-1 (x) + cosec-1 (x) = π / 2

**Things to Remember**

- The secant function is determined by dividing the hypotenuse and adjacent side in a right-angled triangle. It is abbreviated as “sec” and written as Sec = H/A, where H is the hypotenuse, and A is the adjacent side.
- The sum of one and the tangent square function is the secant square x formula, or we may say that one plus the tangent square function can be summarised as the secant squared function.
- The most common types are two common ways to write the Sec Square X Formula. First, about ‘x’ and then with respect to ‘A.’ sec
^{2}x = 1 + tan^{2}x and sec^{2}A = 1 + tan2 A are two ways to state the formula. - Because the secant ratio is derived from the cosine ratio, the secant formula has a reciprocal formula, sec = (1/cos).
- sec
^{2}x–tan^{2}x = 1 is the Pythagorean formula. This is a really handy equation. The squared connection between sin and cos is analogous.

**Solved Questions**

**Ques. In a right-angled triangle, find the value of sec θ if tan θ is given as 3. (3 Marks)**

**Ans.** Tan θ = 3 is the provided parameter.

The following is the equation:

sec2 x − tan2 x = 1,

that is,

sec2 X = 1 + tan2 X

= 1 + 32 = sec2 X

= 10 = sec2 X

Sec X = √10

As a result, the value of the angle’s secant will be √10.

**Ques. Using a secant formula, get Sec X if Cos x equals 4/5. (2 Marks)**

**Ans. **Knowing, Sec X = 1 / Cos X

Sec X = 1 / (4 / 5)

Sec X = 5 / 4.

As a result, Sec X will have a value of 5 / 4.

**Ques. Calculate the angle’s secant value in a right triangle with a hypotenuse of 13 and an adjacent side of 5. (2 Marks)**

**Ans.** The formula for secant of angle X is as follows:

Hypotenuse / Adjacent Side = Sec X

As a result of swapping the values we obtain,

Sec X = 13 / 5

As a result, the secant’s value is 13 / 5.

**Ques. What is the value of sec square x, if tan x = 2/5? (2 Marks)**

**Ans. **Sec2 x = 1 + tan2 x

= 1 + (2/5)2

= 1 + 4/25

= (25 + 4) / 25

= 29/25

Hence, Sec2 x = 29/25

**Ques. What is the value of Sec square x, if tan x = 1/3? (2 Marks)**

**Ans. **Sec2 x = 1 + tan2 x

= 1 + (4/5)2

= 1 + 1/9

= (10) / 9

Hence, Sec2 x = 10/9

**Ques. Find the value of sec****2****x, if the value of cot x = 2/5 (2 Marks)**

**Ans. **cot x = 2/5

tan x = 1/cot x = 5/2

Sec2 x = 1 + tan2 x

= 1 + (5/2)2

= 1 + 25/4

= 29/4

**Ques. Simplify: (1 + tan2A) (1 – sin2A) (2 Marks)**

**Ans. **= (1 + tan2 A)(1 – sin2 A)

= sec2 A . cos2 A

= (1/cos2 A) . cos2 A

= 1

**Ques. What is the value of Sec square x, if tan x = 4/5? (2 Marks)**

**Ans. **Sec2 x = 1 + tan2 x

= 1 + (4/5)2

= 1 + 16/25

= (41) / 25

Sec2 x = 41/25

**Ques. If cot x = ?, then find the value of sec****2****x. (2 Marks)**

**Ans.** cot x = ?

tan x = 1/cot x = 3/2

Sec2 x = 1 + tan2 x

= 1 + (3/2)2

= 1 + 9/4

= 13/4

**Ques. Simplify (1 + tan2A) (1 – sin A) (1 + sin A). (2 Marks)**

**Ans.** (1 + tan2 A) (1 – sin A) (1 + sin A)

= (1 + tan2 A)(1 – sin2 A)

= sec2 A . cos2 A

= (1/cos2 A) . cos2 A

= 1

**Ques. Calculate the length of a right-angled triangle with a 20-unit hypotenuse and a 30-degree base angle with the side. (3 Marks)**

**Ans. **θ = 30 degree

H = 20 units

Sec θ = H/B

Sec 30 =20/B

2 / √3 = 20/B

B = (20 x √3) / 2

B = 10√3

As a result, a right-angle triangle’s base side is 10√3 Units.

**Ques. If the hypotenuse is 13 units, the triangle base is 5 units, and the perpendicular is 12 units, get sec θ using the secant formula. (2 Marks)**

**Ans. **P = 12, B = 5, and H = 13

Sec θ = H / B

sec θ = 13 / 5

sec θ = 2.6

Therefore, sec θ is 2.6

**Ques. Using a secant formula, get Sec θ if Cos θ is given as 3 / 5.** ** (2 Marks)**

**Ans.** Cos θ = 3 / 5

sec θ = (1 / cos θ)

sec θ = 1 / (3 / 5)

sec θ = 5 / 3

Therefore, sec θ is 5 / 3.

**Ques. Prove the Following Identities (Sec X Sec Y + Tan X Tan Y )**^{2} **− ( Sec X Tan Y + Tan X Sec Y )**^{2}** = 1 (5 Marks)**

**Ans. **LHS = (Sec X Sec Y + Tan X Tan Y )^{2} − ( Sec X Tan Y + Tan X Sec Y )^{2}

= [(sec x sec y)2 + (tan x tan y)^{2} – 2(sec x sec y)(tan y x tan y)] – [(sec x tan y)2 + (tan x sec y)^{2} – 2(sec x tan y)(tan y x tan y)]

= [sec^{2} x sec^{2} y + tan^{2} x tan^{2} y – 2sec x sec y tan y x tan y] – [sec^{2} x tan^{2} y + tan^{2} x sec^{2} y – 2sec x tan y tan y x tan y]

= sec^{2}x sec^{2} y + tan^{2}x tan^{2} y – 2sec x sec y tan y x tan y – sec^{2} x tan^{2}y + tan^{2} x sec^{2} y + 2sec x tan y tan y x tan y

= sec^{2}x sec^{2} y + tan^{2}x tan^{2}y – sec^{2}x tan^{2}y + tan^{2} x sec^{2} y

= sec^{2}x(sec^{2} y – tan^{2}y) + tan^{2}x(tan^{2} y – sec^{2} y)

= sec^{2}x(sec^{2} y – tan^{2}y) – tan^{2} x(sec^{2} y – tan^{2} y)

= sec^{2}x(1) – tan^{2} x(1)

= sec^{2} x – tan^{2} x

= 1

= RHS

Since, LHS=RHS, Hence proved.

## Derivative of Sec Square x

The derivative of sec square x is equal to twice the product of sec square x and tanx. Mathematically, we can write the **derivative of sec^2x** as d(sec^2x)/dx = 2 sec^{2}x tanx. We can evaluate the derivative of sec^2x using different methods of differentiation including the chain rule method, product rule and quotient rule of derivatives, and the first principle of derivatives, that is, the definition of limits. Differentiation of a function gives the slope function of the curve of the function which can give the slope of the function at a particular point.

Let us learn to evaluate the derivative of sec square x using different methods of derivatives and its formula. We will also solve different examples related to the concept for a better understanding of the concept of derivatives.

## What is Derivative of Sec^2x?

The derivative of sec^2x is equal to 2 sec^{2}x tanx. It is mathematically written as d(sec^2x)/dx = 2 sec^{2}x tanx. Sec x is one of the important and main trigonometric functions in trigonometry. We know that the derivative of secx is equal to secx tanx. We can use the power rule of differentiation and the derivative of sec x to find the derivative of sec square x. In the next section, let us go through the formula for the differentiation of sec^2x.

## Derivative of Sec Square x Formula

The formula for the derivative of sec^2x is given by d(sec^2x)/dx = 2 sec^2x tanx. We know that the power rule of differentiation is d(x^n)/dx = n x^(n-1), that is, d(x^{n})/dx = nx^{n-1}. Using this formula, we can write the formula for the differentiation of sec square x as d(sec^2x)/dx = 2 secx × d(secx)/dx = 2 secx × secx tanx = 2 sec^{2}x tanx. The image below shows the formula for the derivative of sec square x:

## Derivative of Sec^2x Using Chain Rule

The chain rule of derivatives is used to find the derivatives of composite functions. Sec^2x is the composite function of two functions f(x) = x^{2} and g(x) = secx given by, h(x) = f o g (x) = sec^2x. Now, to find the derivative of sec^2x, we use the formula of chain rule given by, h'(x) = [f(g(x))]’ = f'(g(x)) × g'(x). Hence, using this formula, the power rule of derivatives and the derivative of secx, we have

d(sec^2x)/dx = 2 sec^{2-1}x × d(secx)/dx

= 2 secx × secx tanx

= 2 sec^{2}x tanx

Hence, the derivative of sec square x is equal to 2 sec^{2}x tanx.

## Differentiation of Sec Square Using Quotient Rule

Now that we know that the derivative of sec^2x is equal to 2 sec^{2}x tanx, we will prove this using the quotient rule of derivatives. We know that sec x is the reciprocal identity of cos x, that is, secx = 1/cosx. Also, the formula for the quotient rule of differentiation is (f/g)’ = (f’g – fg’)/g^{2}. Using these formulas, we have

d(sec^{2}x)/dx = d(1/cos^{2}x)/dx

= (1/cos^{2}x)’

= [(1)’ cos^{2}x – 1 (cos^{2}x)’]/cos^{4}x

= (0 × cos^{2}x + 2 cosx sinx)/cos^{4}x — [Because the derivative of cosx is equal to -sinx]

= (2 cosx sinx)/cos^{4}x

= 2 sinx/cos^{3}x

= 2 (1/cos^{2}x) (sinx/cosx)

= 2 sec^{2}x tanx

Hence, we have proved that the derivative of sec square is equal to 2 sec^{2}x tanx.

## Derivative of Sec^2x Using Product Rule

The product rule of differentiation is used to find the derivative of the product of two functions. For a function h(x) = f(x) g(x), the formula for its derivative is given by h'(x) = f'(x) g(x) + f(x) g'(x). For the function sec square x, we can writ it as h(x) = sec^{2}x = secx × secx. Therefore, to find the derivative of sec^2x, we have

d(sec^{2}x)/dx = (secx × secx)’

= (secx)’ secx + secx (secx)’

= secx tanx secx + secx secx tanx — [Because the derivative of secx is equal to secx tanx]

= 2 sec^{2}x tanx

Hence, the derivative of sec^2x is equal to 2 sec^{2}x tanx.

**Important Notes on Derivative of Sec Square x**

- The derivative of sec^2x is equal to 2 sec
^{2}x tanx. - The formula for the derivative of sec square x is d(sec
^{2}x)/dx = 2 sec^{2}x tanx - We can find the differentiation of sec square x using different methods of differentiation such as product rule, quotient rule, and chain rule.

## Derivative of Sec Square x Examples

**Example 1:** Determine the derivative of sec square x square with respect to x square.

**Solution: **To find the derivative of sec square x square with respect to x square, that is, d(sec^{2}(x^{2}))/d(x^{2}), we will use the chain rule method of derivatives.

Assume u = sec^{2}(x^{2}) and v = x^{2}. First, find the value of du/dx and dv/dx.

du/dx = d(sec^{2}(x^{2}))/dx

= 2 sec(x^{2}) × sec(x^{2}) tan(x^{2}) × 2x

= 4x sec^{2}(x^{2}) tan(x^{2})

dv/dx = d(x^{2})/dx

= 2x

To find the derivative of sec square x square with respect to x square, we need to find the value of du/dv. Therefore, we have

d(sec^{2}(x^{2}))/d(x^{2}) = du/dv

= [4x sec^{2}(x^{2}) tan(x^{2})]/2x

= 2 sec^{2}(x^{2}) tan(x^{2})

**Answer:** The derivative of sec^{2}(x^{2}) w.r.t. x^{2} is equal to 2 sec^{2}(x^{2}) tan(x^{2}).

**Example 2:** What is the second derivative of sec^2x?

**Solution:** To find the second derivative of sec^2x, we will differentiate its first derivative. We know that the derivative of sec^2x is equal to 2 sec^{2}x tanx. Differentiating this, we have

d^{2}(sec^{2}x)/dx^{2} = d(2 sec^{2}x tanx)/dx

= 2 [(sec^{2}x)’ tanx + sec^{2}x (tanx)’] — [Using product rule of differentiation]

= 2 [2 sec^{2}x tanx × tanx + sec^{2}x × sec^{2}x] — [Because the derivative of sec square x is 2 sec^{2}x tanx and the derivative of tanx is sec^{2}x]

= 2 [2 sec^{2}x tan^{2}x + sec^{4}x]

= 2 sec^{2}x [2 tan^{2}x + sec^{2}x]

**Answer:** The second derivative of sec square x is 2 sec^{2}x [2 tan^{2}x + sec^{2}x].

**Example 3:** Prove that the derivative of sec square x is 2 sec^{2}x tanx using the trigonometric identity 1 + tan^{2}x = sec^{2}x.

**Solution:** We know that 1 + tan^{2}x = sec^{2}x. Now, replacing sec^{2}x with 1 + tan^{2}x in the derivative, we have

d(sec^{2}x)/dx = d(1 + tan^{2}x)/dx

= d(1)/dx + d(tan^{2}x)/dx

= 0 + 2 tanx sec^{2}x — [Because derivative of tanx is sec^{2}x and d(x^{n})/dx = nx^{n-1}]

= 2 sec^{2}x tanx

**Answer:** Hence Proved.

## FAQs on Derivative of Sec^2x

### What is Derivative of Sec Square x?

The **derivative of sec square x** is equal to twice the product of sec square x and tanx. Mathematically, we can write the derivative of sec^2x as d(sec^2x)/dx = 2 sec^{2}x tanx.

### What is the Formula for the Derivative of Sec^2x?

The formula for the derivative of sec^2x is given by d(sec^{2}x)/dx = 2 sec^{2}x tanx. Using the formula for the power rule of derivatives, we can write the formula for the differentiation of sec square x as d(sec^2x)/dx = 2 secx × d(secx)/dx = 2 secx × secx tanx = 2 sec^{2}x tanx.

### How Do You Prove the Derivative of Sec Square x?

We can evaluate the derivative of sec^2x using different methods of differentiation including the chain rule method, product rule and quotient rule of derivatives, and the first principle of derivatives, that is, the definition of limits.

### What is the Derivative of Sec Square x Square w.r.t. x Square?

The derivative of sec square x square with respect to x square is equal to 2 sec^{2}(x^{2}) tan(x^{2}). We can find this using the chain rule method of differentiation.

### How to Find the Second Derivative of Sec^2x?

To find the second derivative of sec^2x, we will differentiate its first derivative. We know that the derivative of sec^2x is equal to 2 sec^{2}x tanx. Differentiating this gives us the second derivative of sec square x. The second derivative of sec square x is 2 sec^{2}x [2 tan^{2}x + sec^{2}x].

### What is the Antiderivative of Sec^2x?

The antiderivative of sec^2x is equal to tanx + C, where C is the integration constant. The antiderivative of a function is nothing but its integral and we know that the derivative of tanx is equal to sec^{2}x. Therefore, the antiderivative of sec square x is tanx + C.

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