Question

Prove that $\frac{1+\sin \theta }{\cos \theta }+\frac{\cos \theta }{1+\sin \theta }=2\sec \theta$

Answer 1

$LHS=\frac{1+\sin \theta }{\cos \theta }+\frac{\cos \theta }{1+\sin \theta }$

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

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

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

$=2\sec \theta$

$=RHS$

Hence proved