a) \( \frac{\sin ^{2} x+\sin ^{2} x \cos x}{1+\cos x} \)

To simplify the expression \( \frac{\sin^{2} x + \sin^{2} x \cos x}{1 + \cos x} \), we can factor out \( \sin^{2} x \) from the numerator: \[ = \frac{\sin^{2} x (1 + \cos x)}{1 + \cos x} \] Next, as long as \( 1 + \cos x \neq 0 \) (which happens when \( x \) is an odd multiple of \( \pi \)), we can cancel \( 1 + \cos x \): \[ = \sin^{2} x \] Thus, for \( x \) not equal to those odd multiples, the expression simplifies to: \[ \sin^{2} x \] For the special cases when \( x = (2n+1)\pi \) where \( n \) is an integer, we would need to evaluate \( \frac{\sin^{2} x}{0} \), which tends to be undefined. So the simplified form is valid except at these specific points!