First slide

Multiple and sub- multiple Angles

Question

For $α = π / 7$ which of the following hold(s) good?

Moderate

A

$\tan αtan 2 αtan 3 α = \tan 3 α - \tan 2 α - \tan α$
B

$cosec α = cosec 2 α + cosec 4 α$
C

$\cos α - \cos 2 α + \cos 3 α = 1 / 2$
D

$8 \cos αcos 2 αcos 4 α = 1$

Solution

(1) $\tan αtan 2 αtan 3 α = \tan 3 α - \tan 2 α - \tan α$
always holds good. $(∵ \tan 2 α = \tan (3 α - α))$
$\begin{matrix} (2) R.H.S. = \frac{\sin 4 α + \sin 2 α}{\sin 2 αsin 4 α} \\ = \frac{2 \sin 3 αcos α}{\sin 2 αsin 4 α} = \frac{2 \sin 4 α \cdot \cos α}{\sin 2 α \cdot \sin 4 α} \\ = \frac{1}{\sin α} = cosec α (using π / 7 = α) \end{matrix}$
Hence, (2) is correct.
$\begin{matrix} Also \cos 2 α = \cos \frac{2 π}{7} = - \cos (π - \frac{5 π}{7}) \\ = - \cos (\frac{5 π}{7}) = - \cos 5 α \end{matrix}$
$\begin{matrix} (3) \cos α + \cos 3 α + \cos 5 α = \frac{\sin 3 α}{\sin α} \cos 3 α \\ = \frac{\sin 6 α}{2 \sin α} = \frac{\sin \frac{6 π}{7}}{2 \sin \frac{π}{7}} \\ = \frac{\sin (π - \frac{π}{7})}{2 \sin \frac{π}{7}} = \frac{1}{2} \end{matrix}$
$(4) 8 \cos αcos 2 αcos 4 α = \frac{\sin 8 α}{\sin α} = \frac{\sin \frac{8 π}{7}}{\sin \frac{π}{7}} = - \frac{\sin \frac{π}{7}}{\sin \frac{π}{7}} = - 1$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App