Ctga/tga+1=1/sin²a доказать тождество

$\frac{ctg \left(a\right)}{tg \left(a\right)}+1=\frac{1}{\sin ^2\left(a\right)}\\\\1+\frac{\frac{\cos \left(a\right)}{\sin \left(a\right)}}{\frac{\sin \left(a\right)}{\cos \left(a\right)}}=\frac{1}{\sin ^2\left(a\right)}\\\\\frac{\cos ^2\left(a\right)+\sin ^2\left(a\right)}{\sin ^2\left(a\right)}=\frac{1}{\sin ^2\left(a\right)}\\\\\frac{1}{\sin ^2\left(a\right)}=\frac{1}{\sin ^2\left(a\right)}$

тождество доказано