Question

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

`(tan theta)/(1-cot theta) + (cot theta)/(1-tan theta) = 1+secthetacosectheta`

[Hint: Write the expression in terms of sinθ and cosθ]

Answer 1

L.H.S

= `(tantheta)/(1-cottheta) + (cottheta)/(1-tantheta) `

= `(sintheta/costheta)/(1-costheta/sintheta) + (costheta/sintheta)/(1-sintheta/costheta)`

= `(sintheta/costheta)/((sintheta-costheta)/(sintheta))+ (costheta/sintheta)/((costheta-sintheta)/costheta)`

= `(sin^2theta)/(costheta(sintheta-costheta)) - (cos^2theta)/(sintheta(sintheta-costheta))`

= `1/(sintheta - costheta)[(sin^2theta)/costheta - cos^2theta/sintheta]`

= `(1/(sintheta-costheta))[(sin^3theta-cos^3theta)/(sinthetacostheta)]`

= `(1/(sintheta-costheta))[((sintheta-costheta)(sin^2theta+cos^2theta+sinthetacostheta))/(sinthetacostheta)]`

= `((1+sinthetacostheta))/((sinthetacostheta))`

= sec θ cosec θ + 1

= R.H.S