Trig1.13

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

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

Simplifying,we get

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

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

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

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

$\frac{2}{\cos\theta}=2\sec\theta\,$ =RHS