If the sum of all possible values of x satisfying sin 3 xsin 3 x-cos x-sin xsin x-cos 3 x- where x -0-2 - is a -b- where a and b are coprime- then the value of a - b is

Given : sin 3x (sin 3x cos x) = sin x (sin x cos 3x) ⇒sin^2 3x sin^2 x = sin 3xcos x sin xcos 3x ⇒ nbsp;(sin 3x sin x)(sin 3x+sin x)=sin 2x ⇒sin 2 xsin ...

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Byju's Answer

Standard XII

Mathematics

Sum of Trigonometric Ratios in Terms of Their Product

If the sum of...

Question

If the sum of all possible value(s) of $x$ satisfying $sin 3 x (sin 3 x - cos x) = sin x (sin x - cos 3 x)$ , where $x \in [0, 2 π]$ is $\frac{a π}{b}$ , where $a$ and $b$ are coprime, then the value of $(a + b)$ is

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Solution

Given :
$sin 3 x (sin 3 x - cos x) = sin x (sin x - cos 3 x) \Rightarrow {sin}^{2} 3 x - {sin}^{2} x = sin 3 x cos x - sin x cos 3 x \Rightarrow (sin 3 x - sin x) (sin 3 x + sin x) = sin 2 x \Rightarrow sin 2 x sin 4 x = sin 2 x \Rightarrow sin 2 x (sin 4 x - 1) = 0 \Rightarrow sin 2 x = 0 or sin 4 x = 1$

When $sin 2 x = 0$ , we get
$2 x = 0, π, 2 π, 3 π, 4 π \Rightarrow x = 0, \frac{π}{2}, π, \frac{3 π}{2}, 2 π$

When $sin 4 x = 1$ , we get
$4 x = \frac{π}{2}, \frac{5 π}{2}, \frac{9 π}{2}, \frac{13 π}{2} \Rightarrow x = \frac{π}{8}, \frac{5 π}{8}, \frac{9 π}{8}, \frac{13 π}{8}$

Therefore, the sum of all possible values of $x$ is
$5 π + \frac{7 π}{2} = \frac{17 π}{2} = \frac{a π}{b} ∴ a + b = 19$

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If the sum of all possible value(s) of x satisfying sin3x(sin3x−cosx)=sinx(sinx−cos3x), where x∈[0,2π] is aπb, where a and b are coprime, then the value of (a+b) is

If the sum of all possible value(s) of $x$ satisfying $sin 3 x (sin 3 x - cos x) = sin x (sin x - cos 3 x)$ , where $x \in [0, 2 π]$ is $\frac{a π}{b}$ , where $a$ and $b$ are coprime, then the value of $(a + b)$ is