Question

Using the identity ${\sin }^{2}A+{\cos }^{2}A=1$,prove that $\frac{1+\sin \theta }{\cos \theta }+\frac{\cos \theta }{1+\sin \theta }=2sec\theta .$

Answer 1

Simplying the expression:

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

Now, $L.H.S.=\frac{1+\sin \theta }{\cos \theta }+\frac{\cos \theta }{1+\sin \theta }$

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

$=\frac{1+2\sin \theta +{\sin }^2\theta +{\cos }^2\theta }{\cos \theta \left(1+\sin \theta \right)}$ $\left[\because {(a+b)}^2=a^2+b^2+2ab\right]$

$=\frac{1+2\sin \theta +1}{\cos \theta \left(1+\sin \theta \right)}$ $\left[\because {\sin }^{2}A+{\cos }^{2}A=1\right]$

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

$=\frac{2}{\cos \theta }$

$=2\text{sec}\theta$

$=R.H.S.$

Hence, it is proved that$\frac{1+\sin \theta }{\cos \theta }+\frac{\cos \theta }{1+\sin \theta }=2sec\theta .$