Question

Prove the following trigonometric identities.

(cosecA − sinA) (secA − cosA) (tanA + cotA) = 1

Answer 1

We have to prove (cosecA − sinA) (secA − cosA) (tanA + cotA) = 1

We know that `sin^2 A + cos^2 A = 1`

So

(cosec A - sin A)(sec A - cos A)(tan A + cot A)

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

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

`= ((cos^2 A)/sin A) ((sin^2 A)/cos A) (1/(sin A cos A))`

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

 = 1