[tex]\bf sin^2(\theta)+cos^2(\theta)=1\qquad \qquad csc(\theta)=\cfrac{1}{sin(\theta)}\\\\
-----------------------------\\\\
\cfrac{1-cos(x)}{sin(x)}+\cfrac{sin(x)}{1-cos(x)}=2csc(x)\\\\
-----------------------------\\\\[/tex]
[tex]\bf \cfrac{1-cos(x)}{sin(x)}+\cfrac{sin(x)}{1-cos(x)}\implies \cfrac{[1-cos(x)]^2+sin^2(x)}{sin(x)[1-cos(x)]}
\\\\\\
\cfrac{1-2cos(x)+\boxed{cos^2(x)+sin^2(x)}}{sin(x)[1-cos(x)]}\implies \cfrac{1-2cos(x)+1}{sin(x)[1-cos(x)]}
\\\\\\
\cfrac{2-2cos(x)}{sin(x)[1-cos(x)]}\implies \cfrac{2\underline{[1-cos(x)]}}{sin(x)\underline{[1-cos(x)]}}
\\\\\\
\cfrac{2}{sin(x)}\implies 2\cdot \cfrac{1}{sin(x)}\implies 2csc(x)[/tex]
