Question

find all the points of intersection: y=cosx and y=1- sin(x/2)

Answer 1

One way to do it:
\[\begin{align*}
\cos x&=1-\sin\frac{x}{2}\\\\
\sin\frac{x}{2}&=1-\cos x\\\\
\left(\sin\frac{x}{2}\right)^2&=(1-\cos x)^2&\text{half-angle identity for next step}\\\\
\frac{1}{2}-\frac{1}{2}\cos x&=1-2\cos x+\cos^2x\\\\
\cos^2x-\frac{3}{2}\cos x-\frac{1}{2}&=0\\\\
2\cos^2x-3\cos x-1&=0\end{align*}\]

Answer 2

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