simplify: \(\frac{1 +cos\theta}{sin\theta} +\frac{sin\theta}{1+cos\theta}\)

Given:
$\frac{1+\cos\theta}{\sin\theta} + \frac{\sin\theta}{1+\cos\theta}$
Taking LCM:
$= \frac{(1+\cos\theta)^2+\sin^2\theta}{\sin\theta(1+\cos\theta)}$
Expand numerator:
$= \frac{1+2\cos\theta+\cos^2\theta+\sin^2\theta}{\sin\theta(1+\cos\theta)}$
Using identity:
$\sin^2\theta+\cos^2\theta=1$
$= \frac{1+2\cos\theta+1}{\sin\theta(1+\cos\theta)}$
$= \frac{2(1+\cos\theta)}{\sin\theta(1+\cos\theta)}$
$= \frac{2}{\sin\theta}$
$= 2\cosec\theta$
Correct Answer: $2\cosec\theta$