To prove cos3theta +2cos5theta +cos7theta/costheta +2cos3theta +cos5theta=cos2theta-sin2theta×tan3theta ?

Jul 22, 2018

Please refer to a Proof in Explanation.

Explanation:

$\text{The Expression} = \frac{\cos 3 \theta + 2 \cos 5 \theta + \cos 7 \theta}{\cos \theta + 2 \cos 3 \theta + \cos 5 \theta}$.

$\text{The Nr.} = \left(\cos 3 \theta + 2 \cos 5 \theta + \cos 7 \theta\right)$

$= \left(\cos 7 \theta + \cos 3 \theta\right) + 2 \cos 5 \theta$,

$= 2 \cos \left(\frac{7 \theta + 3 \theta}{2}\right) \cos \left(\frac{7 \theta - 3 \theta}{2}\right) + 2 \cos 5 \theta$,

$= 2 \cos 5 \theta \cos 2 \theta + 2 \cos 5 \theta$,

$\therefore \text{ The Nr.} = 2 \cos 5 \theta \left(\cos 2 \theta + 1\right)$.

$\text{On similar lines the Dr.} = 2 \cos 3 \theta \left(\cos 2 \theta + 1\right)$.

$\therefore \text{The Exp.} = \frac{\cos 5 \theta}{\cos 3 \theta}$,

$= \frac{\cos \left(3 \theta + 2 \theta\right)}{\cos 3 \theta}$,

$= \frac{\cos 3 \theta \cos 2 \theta - \sin 3 \theta \sin 2 \theta}{\cos 3 \theta}$,

$= \frac{\cos 3 \theta \cos 2 \theta}{\cos 3 \theta} - \frac{\sin 3 \theta \sin 2 \theta}{\cos 3 \theta}$,

$= \cos 2 \theta - \left(\sin 2 \theta\right) \left\{\frac{\sin 3 \theta}{\cos 3 \theta}\right\}$,

$= \cos 2 \theta - \sin 2 \theta \cdot \tan 3 \theta$, as desired!