The value of the expression $\frac{sin 2 x cos 3 x + sin 3 x cos 8 x + sin 5 x cos 16 x}{sin 2 x sin 3 x + sin 3 x sin 8 x + sin 5 x sin 16 x}$ is

cot 11 x

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tan 11 x

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cot 10 x

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tan 10 x

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Solution

The correct option is A $cot 11 x$
$sin 2 x cos 3 x + sin 3 x cos 8 x + sin 5 x cos 16 x$
$= \frac{1}{2} [(sin 5 x - sin x) + (sin 11 x - sin 5 x) + (sin 21 x - sin 11 x)]$
$= \frac{1}{2} [sin 21 x - sin x] = cos 11 x sin 10 x$

$sin 2 x sin 3 x + sin 3 x sin 8 x + sin 5 x sin 16 x$
$= \frac{1}{2} [(cos x - cos 5 x) + (cos 5 x - cos 11 x) + (cos 11 x - cos 21 x)]$
$= \frac{1}{2} [cos x - cos 21 x] = sin 11 x sin 10 x$

$∴ \frac{sin 2 x cos 3 x + sin 3 x cos 8 x + sin 5 x cos 16 x}{sin 2 x sin 3 x + sin 3 x sin 8 x + sin 5 x sin 16 x} = cot 11 x$

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