Question

Prove the following trigonometric identities:

(1+cotA - cosecA)(1 + tanA + secA) = 2

Answer 1

= \(\Big(1+\frac{cosθ}{sinθ}-\frac{1}{sinθ}\Big)\) \(\Big(1+\frac{cosθ}{sinθ}+\frac{1}{sinθ}\Big)\) 

= \(\Big(\frac{sinθ+cosθ-1}{sinθ}\Big)\) \(\Big(\frac{cosθ+sinθ+1}{cosθ}\Big)\) 

= \(\frac{[(sinθ+cosθ)-1][(sinθ+cosθ)+1]}{sinθ.cosθ}\) 

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

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

= \(\frac{1+2sinθcosθ-1}{sinθ.cosθ}\) 

= \(\frac{2sinθcosθ}{sinθ.cosθ}\) = 2 = R.H.S

Hence Proved.