Cosec Cot Formula

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Cosec Cot

Trigonometry is the field of study which deals with the relationship between angles, heights, and lengths of right triangles. And this time we will be covering Cosec Cot Formula. The ratios of the sides of a right triangle are known as trigonometric ratios. Trigonometry has six main ratios namely sin, cos, tan, cot, sec, and cosec. All these ratios have different formulas. It uses the three sides and angles of a right-angled triangle. Let’s look into Cosec Cot Formulas in detail.

What Is Cosec Cot Formula?

Let’s look into the Cosec Cot Formula

For an acute angle x in a right triangle, Cosec x is given by

Cosec x = Hypotenuse / Opposite side

Cot x is given by,

Cot x = Adjacent Side / Opposite Side

The Cosec Cot Formula is given as follows:

1+cot2θ=cosec2θ

Sec, Cosec and Cot

Secant, cosecant and cotangent, almost always written as sec, cosec and cot are trigonometric functions like sin, cos and tan.

Graphs of sec x and cosec x

Examples using Cosec Cot Formula

Example 1: Prove that (cosec θ – cot θ)2 = (1 – cos θ)/(1 + cos θ)

Solution

LHS = (cosec θ – cot θ)2

= (1/sin θ−cosθ/sin θ)2

= ((1−cos θ)/sin θ)2

RHS = (1 – cos θ)/(1 + cos θ)

By rationalising the denominator,

= (1−cos θ)/(1+cos θ)×(1−cos θ)(1−cos θ)

= (1−cos θ)2/(1−cos2θ)

= (1−cos θ)2/sin2θ

= ((1−cos θ)/sin θ)2

Therefore, LHS = RHS

Example 2: Find Cot P if Tan P = 4 / 3

Solution: 

Using Cotangent formula we know that,

Cot P = 1 / Tan P

= 1 / (4 / 3)

= 3/4

Thus, Cot P = 3/4

Examples of Cosec Cot formula

Q1) If Sin x = ⅗, find the value of Cosec x?

Cosec x = 1/sinx

= 5/3.

Q2) If Sin x = 5/7, find the value of Cosec x?

Cosec x = 1/sinx

= 7/5

Q.3: Prove that (cosec θ – cot θ)2 = (1 – cos θ)/(1 + cos θ).

Solution:
LHS = (cosec θ – cot θ)2

Sec, Cosec and Cot

Summary

Recall, in case of a right angle triangle if we are given one length and one angle and we have to find a missing length or if we need to find a missing angle when two lengths are given we use SOH/CAH/TOA where:

SOH

CAH

TOA

Graphical Comparison

Example 1

Q. In the triangle, find cosec⁡(A)sec⁡(A), and cot⁡(A).

Cosecant, Secant & Cotangent

Tangent, Cotangent, Secant, and Cosecant

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