Prove the following trigonometric identities: (1+tan^2θ)cotθ/cosec^2θ =tanθ

Question

Prove the following trigonometric identities:

\(\frac{(1+tan^2θ)cotθ}{cosec^2θ}\) = tanθ

Answer 1

\(\frac{(1+tan^2θ)cotθ}{cosec^2θ}\)   =   \(\frac{sec^2\times θcotθ}{cosec^2θ}\)

=  \(\frac{sin^2θ}{cos^2θ}\times cotθ\)

 = tan²θ x cotθ

 = tan²θ x \(\frac{1}{tanθ}\)

 = tanθ 

Hence Proved.