If \(y = \displaystyle\frac{\sin x}{1 + \cos x}\), \(-\pi \lt x \lt \pi\), then \(\displaystyle\frac{dy}{dx}\) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{1}{1 + \cos x}\)
By the quotient rule:
\( \displaystyle \frac{dy}{dx} = \frac{\cos x(1 + \cos x) - \sin x(-\sin x)}{(1 + \cos x)^{2}} \)
\( \displaystyle = \frac{\cos x + \cos^{2} x + \sin^{2} x}{(1 + \cos x)^{2}} \)
Since \(\cos^{2} x + \sin^{2} x = 1\):
\( \displaystyle \frac{dy}{dx} = \frac{1 + \cos x}{(1 + \cos x)^{2}} = \frac{1}{1 + \cos x} \)