Question

Prove the following trigonometric identities:

(cosec  θ + sin θ)(cosecθ - sin θ) = cot²θ + cos²θ

Answer 1

Consider, 

(cosecθ + sinθ)(cosecθ - sinθ) 

Apply the formula (a² - b²) = (a+b)(a-b) 

we get, 

(cosecθ + sinθ)(cosecθ - sinθ) = cosec²θ - sin²θ 

As we know 1+cot²A = cosec²A 

and 1-cos²A = sin²A 

So, 

(cosecθ + sinθ)(cosecθ - sinθ) 

= (1+cot²A) -(1-cos²A ) 

= 1+cot²A - 1+cos²A 

= cot²A +cos²A 

Hence Proved