If cosecθ + cotθ = p, then prove that cosθ = p² -1 / p²+1

Solution:

Given, cosecθ + cotθ = p ----------- (1)

We have to prove that cosθ = p² - 1 / p² + 1

Using trigonometric identities,

cot² A + 1 = cosec² A

So, cosec² A - cot² A = 1

By using algebraic identity,

(a² - b²) = (a + b)(a - b)

(cosec θ - cot θ)(cosec θ + cot θ) = 1

(cosec θ - cot θ)(P) = 1

cosec θ - cot θ = 1/P ---------------------- (2)

Adding (1) and (2),

cosec θ - cot θ + cosec θ + cot θ = P + 1/P

2cosec θ = P + 1/P

cosec θ = (P + 1/P)/2

Subtracting (1) and (2),

cosec θ + cot θ - cosec θ + cot θ = P - 1/P

2cot θ = (P - 1/P)

cot θ = (P - 1/P)/2 -------------------------- (3)

Dividing (3) by (2),

cot θ / cosec θ = (P - 1/P)/2 / (P + 1/P)/2

cot θ / cosec θ = (P - 1/P) / (P + 1/P)

We know, cot A = cos A/sin A

Also, cosec A = 1/sin A

Now, cot θ/cosec θ = (cos θ/sin θ) / (1/sin θ)

= (cos θ/sin θ)(sin θ)

= cos θ

So, cos θ = (P - 1/P) / (P + 1/P)

= (P² - 1)/P / (P² + 1)/P

= (P² - 1)/(P² + 1)

Therefore, cos θ = (P² - 1)/(P² + 1)

✦ Try This: Prove the following identities: (cosθ - sinθ + 1)/(cosθ + sinθ - 1) = cosecθ + cotθ

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 8

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NCERT Exemplar Class 10 Maths Exercise 8.4 Problem 1

If cosecθ + cotθ = p, then prove that cosθ = p² -1 / p²+1

Summary:

Trigonometry is the branch of mathematics that deals with the relationship between ratios of the sides of a right-angled triangle with its angles. If cosecθ + cotθ = p. It is proven that cosθ = p² -1 / p²+1

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