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Question Number 129231 by liberty last updated on 14/Jan/21

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 \dots

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