Let be such that cot x = - 5 11 . Then ( sin 11 x 2 ) ( sin 6 x - cos 6 x ) + ( cos 11 x 2 ) ( sin 6 x + cos 6 x ) is equal to [2024]

Question

Let $\frac{π}{2} < x < π$ be such that $\cot x = \frac{- 5}{\sqrt{11}}$ . Then

$(\sin \frac{11 x}{2}) (\sin 6 x - \cos 6 x) + (\cos \frac{11 x}{2}) (\sin 6 x + \cos 6 x)$ is equal to [2024]

Smart Achievers · Accepted Answer

(2)

Let $E = (\sin 6 x \cos \frac{11 x}{2} - \cos 6 x \sin \frac{11 x}{2}) + (\cos 6 x \cos \frac{11 x}{2} + \sin 6 x \sin \frac{11 x}{2})$

$= (\sin (6 x - \frac{11 x}{2}) + \cos (6 x - \frac{11 x}{2}))$

$= \sin \frac{x}{2} + \cos \frac{x}{2}$

$Now, E^{2} = 1 + \sin x \dots (i)$

$∴$ $\cot x = \frac{- 5}{\sqrt{11}}$

$∴$ $E^{2} = 1 + \frac{\sqrt{11}}{6}$

$∴$ $E = \sqrt{\frac{6 + \sqrt{11}}{6}} = \sqrt{\frac{12 + 2 \sqrt{11}}{12}} = \frac{\sqrt{11} + 1}{2 \sqrt{3}}$

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