Question

Prove the following trigonometric identities.

`(1 + cos A)/sin A = sin A/(1 - cos A)`

Answer 1

We need to prove `(1 + cos A)/sin A = sin A/(1 - cos A)`

Now, multiplying the numerator and denominator of LHS by `1 - cos A` we get

`(1 + cos A)/sin A = (1 + cos A)/sin A xx (1 - cos A)/(1 - cos A)`

Further using the identity,  `a^2 - b^2 = (a + b)(a - b)` we get

`(1 + cos A)/sin A xx (1 - cos A)/(1 - cos A) = (1 - cos^2 A)/(sin A (1- cos A))`

`= sin^2 A/(sin A(1 - cos A))`         (Using `sin^2 theta + cos^2 theta = 1`)

`= sin A/(1 - cos A)`

Hence proved