Question

Prove that:

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

Prove the following trigonometric identities:

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

Answer 1

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

Expanding both terms in the product:

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

Using the identity:

1 − cos²A = sin²A, 1 − sin²A = cos²A

`((sin^2A)/(cos A)) xx ((cos^2 A)/(sin A))`

Simplify the equation

`(sin^2 A)/(cos A) xx (cos^2 A)/(sin A)`

Cancel out common terms `(sin A cos A)/(cos A sin A)`

∴ `(sin A cos A)/(cos A sin A) = 1/(tan A + cot A)`

Hence proved.