[tex]\bf sin^2(\theta)+cos^2(\theta)=1\implies cos^2(\theta)=1-sin^2(\theta)
\\\\\\
tan(\theta)=\cfrac{sin(\theta)}{cos(\theta)}
\qquad \qquad
cot(\theta)=\cfrac{cos(\theta)}{sin(\theta)}\\\\
-------------------------------\\\\
\cfrac{cot(x)}{tan(x)+cot(x)}=1-sin^2(x)
\\\\\\
\textit{doing the left-side}\implies \cfrac{\frac{cos(x)}{sin(x)}}{\frac{sin(x)}{cos(x)}+\frac{cos(x)}{sin(x)}}\implies
\cfrac{\frac{cos(x)}{sin(x)}}{\frac{sin^2(x)+cos^2(x)}{cos(x)sin(x)}}[/tex]
[tex]\bf
\cfrac{\frac{cos(x)}{sin(x)}}{\frac{1}{cos(x)sin(x)}}\implies \cfrac{cos(x)}{\underline{sin(x)}}\cdot \cfrac{cos(x)\underline{sin(x)}}{1}\implies cos^2(x)\implies 1-sin^2(x)[/tex]
