Question

Докажите тождество(1+tga^2)cosa^2+sina^2(1+ctga^2)=2

Answer 1

$\cos^2\alpha(1+\mathrm{tg}^2\alpha)+\sin^2\alpha(1+\mathrm{ctg}^2\alpha)=2$

Упростим левую часть выражения

$\cos^2\alpha(1+\mathrm{tg}^2\alpha)+\sin^2\alpha(1+\mathrm{ctg}^2\alpha)=\medskip\\=\cos^2\alpha+\cos^2\alpha~\mathrm{tg}^2\alpha+\sin^2\alpha+\sin^2\alpha~\mathrm{ctg}^2\alpha=\medskip\\=(\cos^2\alpha+\sin^2\alpha)+\left(\dfrac{\cos^2\alpha\sin^2\alpha}{\cos^2\alpha}+\dfrac{\sin^2\alpha\cos^2\alpha}{\sin^2\alpha}\right)=\medskip\\=1+(\sin^2\alpha+\cos^2\alpha)=1+1=2$

$2=2$

Тождество доказано