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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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