Question

Prove that:

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

Answer 1

We have,

L.H.S. = $\frac{(1+sin\theta {)}^2+(1-sin\theta {)}^2}{co{s}^2\theta }$

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

$\Rightarrow$ L.H.S. = $\frac{2+2si{n}^2\theta }{co{s}^2\theta }=\frac{2(1+si{n}^2\theta )}{1-si{n}^2\theta }=2\left(\frac{1+si{n}^2\theta }{1-si{n}^2\theta }\right)=R.H.S.$