Question

The value of
\[ \frac{\sin\theta+\sin3\theta}{\cos\theta+\cos3\theta} \] is

• A. \(\cos2\theta\)
• B. \(\cot2\theta\)
• C. \(\tan2\theta\)
• D. \(\cosec\theta+\sin\theta\)

Hint:

For expressions involving \(\sin A+\sin B\) and \(\cos A+\cos B\), use sum-to-product identities to simplify quickly.

Answer 1

The Correct Option is C

Step 1: Use sum-to-product identities.
We know that
\[ \sin A+\sin B=2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) \]
So,
\[ \sin\theta+\sin3\theta = 2\sin2\theta\cos\theta \]
Also,
\[ \cos A+\cos B=2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) \]
Therefore,
\[ \cos\theta+\cos3\theta = 2\cos2\theta\cos\theta \]

Step 2: Substitute in the given expression.
\[ \frac{\sin\theta+\sin3\theta}{\cos\theta+\cos3\theta} = \frac{2\sin2\theta\cos\theta}{2\cos2\theta\cos\theta} \]
Cancel the common factor \(2\cos\theta\):
\[ =\frac{\sin2\theta}{\cos2\theta} \]
\[ =\tan2\theta \]

Step 3: Final conclusion.
Hence,
\[ \boxed{\tan2\theta} \]