Prove that: (a) \( \frac{\sin x}{1+\cos x}+\frac{1+\cos x}{\sin x}=\frac{2}{\sin x} \)

To prove the given equation: \[ \frac{\sin x}{1+\cos x} + \frac{1+\cos x}{\sin x} = \frac{2}{\sin x} \] we will start by combining the two fractions on the left-hand side. The common denominator of the two fractions is \((1 + \cos x)\sin x\). Thus, we rewrite the left-hand side: \[ \frac{\sin^2 x + (1 + \cos x)^2}{(1 + \cos x)\sin x} \] Now, let's simplify the numerator: \[ (1 + \cos x)^2 = 1 + 2\cos x + \cos^2 x \] So, the numerator now becomes: \[ \sin^2 x + 1 + 2\cos x + \cos^2 x \] Using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), we can substitute: \[ \sin^2 x + \cos^2 x = 1 \] This gives us: \[ 1 + 1 + 2\cos x = 2 + 2\cos x \] Now, we rewrite the full fraction: \[ \frac{2 + 2\cos x}{(1 + \cos x)\sin x} = \frac{2(1 + \cos x)}{(1 + \cos x)\sin x} \] Next, as long as \(1 + \cos x \neq 0\) (which happens at odd multiples of \(\pi\)), we can cancel \(1 + \cos x\) from the numerator and denominator: \[ \frac{2}{\sin x} \] Thus, we have shown that: \[ \frac{\sin x}{1+\cos x} + \frac{1+\cos x}{\sin x} = \frac{2}{\sin x} \] This completes the proof.