Prove the following trigonometric identities: 1+sinθ/cosθ + cosθ/1+sinθ=2sec θ

Question

Prove the following trigonometric identities:

\(\frac{1+sin θ}{cos θ}+\frac{cos θ}{1+sin θ}\) =2sec θ

Answer 1

\(\frac{1+sin θ}{cos θ}+\frac{cos θ}{1+sin θ}\)  = \(\frac{1+sin^2 θ+ 2sin θ+cos^2θ}{cos θ(1+sin θ)}\) 

= \(\frac{1+1+ 2sinθ}{cos θ(1+sin θ)}\) 

=  \(\frac{2+ 2sin θ}{cos θ(1+sin θ)}\)  

= \(\frac{2(1+sin θ)}{cos θ(1+sin θ)}\)  

=  \(\frac{2}{cosθ}\) 

= 2sec θ

Hence Proved