Question

Evaluate \[ \frac{\sin x}{1+\cos x}+\frac{1+\cos x}{\sin x} \]

• A. \(2\sec x\)
• B. \(2\cosec x\)
• C. \(\tan 2x\)
• D. \(\sin 2x\)

Hint:

Whenever an expression contains \(\sin x\) and \(1+\cos x\), try taking the LCM and use the identity \[ \sin^2x+\cos^2x=1 \] to simplify the numerator.

Answer 1

The Correct Option is B

Step 1: Combine over a common denominator.
\[\frac{\sin x}{1+\cos x}+\frac{1+\cos x}{\sin x} = \frac{\sin^2 x + (1+\cos x)^2}{\sin x(1+\cos x)}.\]

Step 2: Simplify the numerator.
\(\sin^2 x + 1 + 2\cos x + \cos^2 x = 2 + 2\cos x = 2(1+\cos x)\). So the expression becomes \(\tfrac{2(1+\cos x)}{\sin x(1+\cos x)} = \tfrac{2}{\sin x} = 2\csc x\).
\[\boxed{2\csc x}\]