If ${sin}^{3} x \cdot sin 3 x = n \sum m = 0 c_{m} \cdot cos m x$ is an identity in x, where $c_{m}$ 's are constants then the value of n is

4

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6

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9

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none of these

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Solution

The correct option is B $6$
Given ${sin}^{3} x sin 3 x = n \sum m = 0 C_{m} cos n x$ is an identity in x,

where $C_{0}, C_{1}, C_{2} . . ., C_{n}$ are constant

${sin}^{3} x sin 3 x = \frac{1}{4} (3 sin x - sin 3 x) sin 3 x$

$= \frac{1}{4} (2 \times \frac{3}{2} sin 3 x sin x - {sin}^{2} 3 x)$

$\Rightarrow n \sum m = 1 (cot (θ + \frac{(m - 1) π}{4}) - cot (θ + \frac{(m) π}{4})) = 4$

$\Rightarrow cot θ - cot (θ + \frac{π}{4}) + cot (θ + \frac{π}{4}) - cot (θ + \frac{2 π}{4}) + cot (θ + \frac{5 π}{4}) - cot (θ + \frac{6 π}{4}) = 4$

$\Rightarrow cot θ - cot (θ + \frac{3 π}{2}) = 4 \Rightarrow cot θ + tan θ = 4 \Rightarrow t a n^{2} θ - 4 tan θ + 1 = 0$

$\Rightarrow (tan θ - 2)^{2} - 3 = 0 \Rightarrow (tan θ - 2 + \sqrt{3}) (tan θ - 2 - \sqrt{3}) = 0$

$\Rightarrow tan θ = 2 - \sqrt{3}$ or $tan θ = 2 + \sqrt{3}$

$\Rightarrow θ = \frac{π}{12}, θ = \frac{5 π}{12} ∵ θ \in (0, \frac{π}{2})$

Suggest Corrections

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