Prove the following trigonometric identities: cot θ - tanθ = 2cos^2 θ - 1/sinθcosθ

Question

Prove the following trigonometric identities:

cot θ - tanθ = \(\frac{2cos^2θ-1}{sinθ.cosθ}\)

Answer 1

L.H.S cot θ - tanθ = \(\frac{cosθ}{sinθ}-\frac{sinθ}{cosθ}\)

 =  \(\frac{cos^2θ-sin^2θ}{sinθ.cosθ}\)   

= \(\frac{cos^2θ-(1-cos^2θ)}{sinθ.cosθ}\)  

= \(\frac{cos^2θ-1+cos^2θ)}{sinθ.cosθ}\)

= \(\frac{2cos^2θ-1}{sinθ.cosθ}\)   

= R.H.S

Hence Proved.