MHB Weekly Math Problem #82: Simplifying Trigonometric Expression

[Jameson]

Simplify $$2 \sin^2 \left(\frac{\pi}{4} - \frac{x}{2} \right)+\sin(x)$$.
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[Jameson]

Congratulations to the following members for their correct solutions:

1) MarkFL
2) anemone
3) Chris L T521
4) kaliprasad
5) soroban

Solution (from soroban):

Spoiler

\text{Simplify: }\:2\sin^2\left(\frac{\pi}{4}-\frac{x}{2}\right) + \sin x

2\left(\sin\tfrac{\pi}{4}\cos\tfrac{x}{2} - \cos\tfrac{\pi}{4}\sin\tfrac{x}{2}\right)^2 + \sin x

=\;2\left(\tfrac{1}{\sqrt{2}}\cos\tfrac{x}{2} - \tfrac{1}{\sqrt{2}}\sin\tfrac{x}{2}\right)^2 + \sin x

=\;2\bigg[\tfrac{1}{\sqrt{2}}\left(\cos\tfrac{x}{2} - \sin\tfrac{x}{2}\right)\bigg]^2 + \sin x

=\;2(\tfrac{1}{2})\left(\cos^2\tfrac{x}{2} - 2\sin\tfrac{x}{2}\cos\tfrac{x}{2} + \sin^2\tfrac{x}{2}\right) + \sin x

=\;\left(1 - 2\sin\tfrac{x}{2}\cos\tfrac{x}{2}\right) + \sin x

=\;1 - \sin x + \sin x

=\;1