Question

Докажите тождества .​

Answer 1

$\displaystyle\bf\\1)\\\\\frac{Cos^{2} \alpha -4Sin^{2} \dfrac{\alpha }{2} Cos^{2} \dfrac{\alpha }{2} }{1-8Sin^{2} \dfrac{\alpha }{2} Cos^{2} \dfrac{\alpha }{2} } =\frac{Cos^{2} \alpha -Sin^{2}\alpha }{1-2Sin^{2} \alpha }=\frac{Cos2\alpha }{Cos2\alpha } =1\\\\\\1=1- \ verno\\\\4)\\\\1-Sin^{2} \Big(\frac{x}{2} -\pi \Big)-Cos^{2} \frac{\pi }{4} +Sin^{2} \frac{\pi }{4} =$

$\displaystyle\bf\\=\underbrace{1-Sin^{2} \frac{x}{2} }_{Cos^{2} \frac{x}{2} } -\Big(\frac{1}{\sqrt{2} } \Big)^{2} +\Big(\frac{1}{\sqrt{2} } \Big)^{2} =Cos^{2} \frac{x}{2} -\frac{1}{2} +\frac{1}{2}=Cos^{2} \frac{x}{2} \\\\\\Cos^{2} \frac{x}{2} =Cos^{2} \frac{x}{2} - \ verno$

Тождество доказано