Question

The value of $\dfrac{\sin\theta + \sin3\theta}{\cos\theta + \cos3\theta}$ is:

• A. $\cos2\theta$
• B. $\cot2\theta$
• C. $\tan2\theta$
• D. $\csc\theta + \sin\theta$

Hint:

Apply sum-to-product identities for sums of sine or cosine of two angles.

Answer 1

The Correct Option is C

Use sum-to-product identities: \[ \sin\theta + \sin3\theta = 2\sin2\theta\cos\theta
\cos\theta + \cos3\theta = 2\cos2\theta\cos\theta \] \[ \Rightarrow \dfrac{\sin\theta + \sin3\theta}{\cos\theta + \cos3\theta} = \dfrac{2\sin2\theta\cos\theta}{2\cos2\theta\cos\theta} = \dfrac{\sin2\theta}{\cos2\theta} = \tan2\theta \]