Question

Решите ур-е. 10 класс
1-cos $\frac{x}{2}$ = tg $\frac{x}{4}$

Answer 1

$1-cos\frac{x}{2}=tg\frac{x}{4}\\\\Formyla:\; sin^2\frac{ \alpha }{2}=\frac{1-cos \alpha }{2}\; \; \to \; \; 1-cos \alpha =2sin^2\frac{ \alpha }{2}\\\\2sin^2\frac{x}{4}=tg\frac{x}{4}\\\\2sin^2\frac{x}{4}-\frac{sin\frac{x}{4}}{cos\frac{x}{4}}=0\; ,\; \; OOF:\; cos\frac{x}{4}\ne 0\\\\2sin^2\frac{x}{4}\cdot cos\frac{x}{4}-sin\frac{x}{4}=0\\\\sin\frac{x}{4}(2sin\frac{x}{4}cos\frac{x}{4}-1)=0\\\\sin\frac{x}{4}(sin\frac{x}{2}-1)=0\\\\a)sin\frac{x}{4}=0,\; \frac{x}{4}=\pi n,\n\in Z\\\\x=4\pi n$

$b)\; sin\frac{x}{2}=1,\; \frac{x}{2}=\frac{\pi}{2}+2\pi k,\; k\in Z\\\\x=\pi +4\pi k$