Given { ((a=sin x+sin 2x)),((b=cos x+cos 2x)) :}. If (a^2 +b^2 )(a^2 +b^2 −3)=cos 2x+3cos x then x =?


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Question Number 129231 by liberty last updated on 14/Jan/21
$Given { ((a=sin x+sin 2x)),((b=cos x+cos 2x)) :}. If (a^2 +b^2 )(a^2 +b^2 −3)=cos 2x+3cos x then x =?$
$Given {\begin{cases} a = \sin x + \sin 2 x \\ b = \cos x + \cos 2 x \end{cases} . If (a^{2} + b^{2}) (a^{2} + b^{2} - 3) = \cos 2 x + 3 \cos x$ $then x = ?$
	Answered by MJS_new last updated on 14/Jan/21

	$let \cos x = c$ $\Rightarrow$ $a = (2 c + 1) \sqrt{1 - c^{2}}$ $b = 2 c^{2} + c - 1$ $a^{2} + b^{2} = 2 c + 2$ $\Rightarrow$ $2 (c + 1) (2 c - 1) = 2 c^{2} + 3 c - 1$ $c^{2} - \frac{1}{2} c - \frac{1}{2} = 0$ $\cos x = - \frac{1}{2} \lor \cos x = 1$ $\Rightarrow$ $x = 2 n π \pm \frac{2 π}{3} \lor x = 2 n π$
	Commented by bemath last updated on 14/Jan/21

	$great . . .$
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