Question

Verify the following identity: ((1-cos(x))/(sin(x)))+((sin(x))/(1-cos(x))) = 2csc(x)

Answer 1

`\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)\\\\ -----------------------------\\\\`


`\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)`