Question

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

Answer 1

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

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

$\frac{1+si{n}^2\theta +2sin\theta +co{s}^2\theta }{cos\theta (1+sin\theta )}=2sec\theta$

$si{n}^2\theta +co{s}^2\theta =1$

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

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

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

$2sec\theta =2sec\theta$ $L.H.S=R.H.S$