First slide

Multiple and sub- multiple Angles

Question

If $θ = 3 α and \sin θ = \frac{a}{\sqrt{a^{2} + b^{2}}},$ the value of the expression $acosec α - bsec α$ is

Moderate

A

$\frac{a}{\sqrt{a^{2} + b^{2}}}$
B

$2 \sqrt{a^{2} + b^{2}}$
C

a+b
D

none of these

Solution

$\begin{array}{l} acosec α - bsec α = \frac{a}{\sin α} - \frac{b}{\cos α} \\ = \frac{\sqrt{a^{2} + b^{2}}}{\sin αcos α} [\frac{a}{\sqrt{a^{2} + b^{2}}} \cos α - \frac{b}{\sqrt{a^{2} + b^{2}}} \sin α] \end{array}$
$Now, \sin 3 α = \frac{a}{\sqrt{a^{2} + b^{2}}} gives$
$\sqrt{a^{2} + b^{2}} [\frac{\sin 3 αcos α - \cos 3 αsin α}{\sin αcos α}] = 2 \sqrt{a^{2} + b^{2}}$

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