Question

$\frac{\sin \theta +\sin 2\theta }{1+\cos \theta +\cos 2\theta }=$

• A. $\frac{1}{2}\tan \theta$
• B. $\frac{1}{2}\cot \theta$
• C. $\tan \theta$
• D. $\cot \theta$

Answer 1

The correct option is C

$\tan \theta$

Explanation for the correct answer:

Simplifying the given expression:

Given $\frac{\sin \theta +\sin 2\theta }{1+\cos \theta +\cos 2\theta }$

We know that,

$$\begin{gathered} \sin 2A=2\sin A\cos A \\ \cos 2A=2{\cos }^{2}A–1 \end{gathered}$$

Applying the above formulas in the given expression, we get

$$\begin{gathered} =\frac{\sin \theta +2\sin \theta \cos \theta }{1+\cos \theta +2{\cos }^2\theta –1} \\ =\frac{\sin \theta \left(1+2\cos \theta \right)}{\cos \theta \left(1+2\cos \theta \right)} \\ =\frac{\sin \theta }{\cos \theta } \\ =\tan \theta \end{gathered}$$

Therefore, $\frac{\sin \theta +\sin 2\theta }{1+\cos \theta +\cos 2\theta }=\tan \theta$

Hence, option (C) is the correct answer.