Solve (√(1+cos x)) + (√(1−cos x)) = 2sin x


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Question Number 134165 by liberty last updated on 28/Feb/21

$Solve \sqrt{1 + \cos x} + \sqrt{1 - \cos x} = 2 \sin x$
	Answered by EDWIN88 last updated on 28/Feb/21
	$square both sides ⇔ 2+2(√(1−cos^2 x)) = 4sin^2 x ⇒ 2+2∣sin x∣ = 4sin^2 x ⇒2sin^2 x ± sin x −1 = 0 ⇒sin x = ((±1 ±3)/4) ∈ {−1,−(1/2),(1/2),2 } since the LHS is always nonnegative so the sine on the right must be nonnegative to. Also sin (x)=(1/2) turns out to be wrong a solution when we try to apply it. This leaves sin (x)=1 ⇒ x=2nπ +(π/2) or x = 2π(n+(1/4))$
	$square both sides$ $\Leftrightarrow 2 + 2 \sqrt{1 - \cos^{2} x} = 4 \sin^{2} x$ $\Rightarrow 2 + 2 ∣ \sin x ∣ = 4 \sin^{2} x$ $\Rightarrow 2 \sin^{2} x \pm \sin x - 1 = 0$ $\Rightarrow \sin x = \frac{\pm 1 \pm 3}{4} \in {- 1, - \frac{1}{2}, \frac{1}{2}, 2}$ $since the LHS is always nonnegative$ $so the sine on the right must be nonnegative$ $to . Also \sin (x) = \frac{1}{2} turns out to be wrong$ $a solution when we try to apply it .$ $This leaves \sin (x) = 1 \Rightarrow x = 2 n π + \frac{π}{2} or$ $x = 2 π (n + \frac{1}{4})$
	Commented by EDWIN88 last updated on 28/Feb/21

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