First slide

Multiple and sub- multiple Angles

Question

$cosec \frac{360^{\circ}}{7} + cosec \frac{540^{\circ}}{7} =$

Moderate

A

$cosec \frac{180^{\circ}}{7}$
B

$cosec \frac{90^{\circ}}{7}$
C

$\sec \frac{180^{\circ}}{7}$
D

$\sec \frac{90^{\circ}}{7}$

Solution

$let α = \frac{180^{\circ}}{7}$
$\begin{matrix} \Rightarrow 180^{\circ} = 7 α \\ \Rightarrow 3 α = 180^{\circ} - 4 α \\ \Rightarrow \sin 3 α = \sin 4 α - - - - (i) \\ ∴ cosec \frac{360^{\circ}}{7} + cosec \frac{540^{\circ}}{7} \\ = cosec 2 α + cosec 3 α \\ = \frac{\sin 3 α + \sin 2 α}{\sin 3 αsin 2 α} \\ = \frac{\sin 4 α + \sin 2 α}{\sin 3 α \cdot \sin 2 α} \\ = \frac{2 \sin 3 αcos α}{\sin 3 α \cdot \sin 2 α} from (i) \\ = \frac{2 \cos α}{2 \sin α \cdot \cos α} \\ = cosec α \end{matrix}$
$= cosec \frac{180^{\circ}}{7}$

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