Question

sin 25x/2+cos 25x/2=1
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Answer 1

$sin\frac{25x}{2}+cos\frac{25x}{2}=1\; |:\sqrt2\\\\\frac{1}{\sqrt2}\cdot sin\frac{25x}{2}+\frac{1}{\sqrt2}\cdot cos\frac{25x}{2}=\frac{1}{\sqrt2}\\\\cos\frac{\pi}{4}\cdot  sin\frac{25x}{2}+sin\frac{\pi}{4}\cdot cos\frac{25x}{2}=\frac{\sqrt2}{2}\\\\sin(\frac{25x}{2}+\frac{\pi}{4})=\frac{\sqrt2}{2}\\\\\frac{25x}{2}+\frac{\pi}{4}=(-1)^{n}\cdot \frac{\pi}{4}+\pi n,\; n\in Z\\\\\frac{25x}{2}=-\frac{\pi}{4}+(-1)^{n}\cdot \frac{\pi}{4}+\pi n\; ,\; n\in Z\\\\x=-\frac{\pi }{50}+(-1)^{n}\cdot \frac{\pi}{50}+\frac{2\pi n}{25}\; ,\; n\in Z$