Suppose sin3-sin 3x-m-0nCm cos -n x is an identity in x-Where C0-C1- Cn are constants and Cn- 0- Then the value of n is

Given sin^3⁡sin 3x=∑_m=0^nC_m cos ⁡n x is an identity in x, Where C_0,C_1,… C_n are constants . sin^3xsin3x=1/4{3sin x sin3x }.sin3x =1/4 ( 3/2.2sin x.sin3x ...

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Byju's Answer

Standard XIII

Mathematics

Solving Simultaneous Trigonometric Equations

Suppose sin3⁡...

Question

Suppose $s i n^{3} s i n 3 x = \sum_{m = 0}^{n} C_{m} c o s n x$ is an identity in x,
Where $C_{0}, C_{1}, \dots C_{n}$ are constants and $C_{n} \neq 0$ . Then the value of n is___

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Solution

Given $s i n^{3} s i n 3 x = \sum_{m = 0}^{n} C_{m} c o s n x$ is an identity in x,
Where $C_{0}, C_{1}, \dots C_{n}$ are constants .
$s i n^{3} x s i n 3 x = \frac{1}{4} {3 s i n x - s i n 3 x} . s i n 3 x$
$= \frac{1}{4} (\frac{3}{2} .2 s i n x . s i n 3 x - s i n^{2} 3 x) = \frac{1}{4} {\frac{3}{2} (c o s 2 x - c o s x) - \frac{1}{2} (1 - c o s 6 x)} = \frac{1}{8} (c o s 6 x + 3 c o s 2 x - 3 c o s x - 1)$
$∴$ On comparing both sides, we get n=6

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Suppose sin3sin3x=∑nm=0Cm cos n x is an identity in x, Where C0,C1,…Cn are constants and Cn≠0. Then the value of n is___

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Suppose $s i n^{3} s i n 3 x = \sum_{m = 0}^{n} C_{m} c o s n x$ is an identity in x,
Where $C_{0}, C_{1}, \dots C_{n}$ are constants and $C_{n} \neq 0$ . Then the value of n is___