Question

Evaluate : $\left(\frac{\sin 3\theta +\sin 5\theta +\sin 7\theta +\sin 9\theta }{\cos 3\theta +\cos 5\theta +\cos 7\theta +\cos 9\theta }\right)$

• A. $\tan \left(3\theta \right)$
• B. $\cot \left(3\theta \right)$
• C. $\tan \left(6\theta \right)$
• D. $\cot \left(6\theta \right)$

Answer 1

The correct option is C

$\tan \left(6\theta \right)$

Explanation for the correct answer:

Simplifying the equations using trigonometric identities:

$$\begin{gathered} \left(\frac{\sin 3\theta +\sin 5\theta +\sin 7\theta +\sin 9\theta }{\cos 3\theta +\cos 5\theta +\cos 7\theta +\cos 9\theta }\right) \\ \mathrm{On}\mathrm{rearranging} \\ \left(\frac{\left(\sin 9\theta +\sin 3\theta \right)+\left(\sin 7\theta +\sin 5\theta \right)}{\left(\cos 9\theta +\cos 3\theta \right)+\left(\cos 7\theta +\cos 5\theta \right)}\right) \end{gathered}$$

Using the formulas $\sin x+\sin y=2\sin \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)$ and $\cos x+\cos y=2\cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)$

$$\begin{gathered} \Rightarrow \frac{\left[\left(2\sin \left(\frac{9\theta +3\theta }{2}\right)\cos \left(\frac{9\theta -3\theta }{2}\right)\right)+\left(2\sin \left(\frac{7\theta +5\theta }{2}\right)\cos \left(\frac{7\theta -5\theta }{2}\right)\right)\right]}{\left[\left(2\cos \left(\frac{9\theta +3\theta }{2}\right)\cos \left(\frac{9\theta -3\theta }{2}\right)\right)+\left(2\cos \left(\frac{7\theta +5\theta }{2}\right)\cos \left(\frac{7\theta -5\theta }{2}\right)\right)\right]} \\ \Rightarrow \frac{2\sin 6\theta \cos 3\theta +2\sin 6\theta \cos \theta }{2\cos 6\theta \cos 3\theta +2\cos 6\theta \cos \theta } \\ \Rightarrow \frac{2\sin 6\theta \left(\cos 3\theta +\cos \theta \right)}{2\cos 6\theta \left(\cos 3\theta +\cos \theta \right)} \\ \Rightarrow \frac{\sin 6\theta }{\cos 6\theta } \\ \Rightarrow \tan 6\theta \left[\because \frac{\sin x}{\cos x}=\tan x\right] \end{gathered}$$

Therefore, the correct answer is option (C).