Question

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

Answer 1

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

$\Rightarrow$ L.H.S. = $\frac{cos\theta (1+sin\theta )+cos\theta (1-sin\theta )}{(1-sin\theta )(1+sin\theta )}$

$\Rightarrow$ L.H.S. = $\frac{cos\theta +cos\theta sin\theta +cos\theta -cos\theta sin\theta }{1-si{n}^2\theta }$

$\Rightarrow$ L.H.S. = $\frac{2cos\theta }{co{s}^2\theta }$ $[\because 1-si{n}^2\theta =co{s}^2\theta ]$

$\Rightarrow$ L.H.S. = $\frac{2}{cos\theta }=2sec\theta =R.H.S.$