Solve-1-cos-x-1-cos-x-2sin-x-

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

\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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