Question

Prove that (cosec A − sin A) (sec A − cos A) = `1/(tan A + cot A)`.

Answer 1

L.H.S. = (cosec A − sin A) (sec A − cos A)

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

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

= `cos^2A/sin A xx sin^2A/cos A`

L.H.S. = cos A × Sin A

and R.H.S. = `1/(tan A+ cot A)`

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

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

= `(sin A cos A)/1`

R.H.S. = sin A × cos A

∴ L.H.S. = R.H.S.

∴ Hence proved.