Question

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:

`(cosec  θ  – cot θ)^2 = (1-cos theta)/(1 + cos theta)`

Answer 1

L.H.S

= `(cosec  θ  – cot θ)^2`

= `(1/sintheta - costheta/sintheta)^2`

= `(1-costheta)^2/(sin^2 theta)`

= `(1-cos theta)^2/(1-cos^2theta)`

= `((1-costheta)(1-costheta))/((1-costheta)(1+cos theta)) `

= `(1-cos theta)/(1+costheta)`

= R.H.S