Question 199577
*[Tex \LARGE \sin(x) + \sin(x)\cot^2(x) = \csc(x)] ... Start with the given equation.



*[Tex \LARGE \sin(x)\left(1 + \cot^2(x)\right) = \csc(x)] ... Factor out the GCF {{{sin(x)}}}



*[Tex \LARGE \sin(x)\csc^2(x) = \csc(x)] ... Replace the terms in the parenthesis with *[Tex \LARGE \csc^2(x)].



Note: *[Tex \LARGE 1 + \cot^2(x)=\csc^2(x)]




*[Tex \LARGE \sin(x)\left(\frac{1}{\sin^2(x)}\right) = \csc(x)] ... Use the identity *[Tex \LARGE \csc(x)=\frac{1}{\sin(x)}]



*[Tex \LARGE \frac{\sin(x)}{\sin^2(x)} = \csc(x)] ... Multiply



*[Tex \LARGE \frac{\sin(x)}{\sin(x)\ast\sin(x)} = \csc(x)] ... Factor



*[Tex \LARGE \frac{1}{\sin(x)} = \csc(x)] ... Cancel out the common terms.



*[Tex \LARGE \csc(x) = \csc(x)] ... Use the identity *[Tex \LARGE \csc(x)=\frac{1}{\sin(x)}]



So we've proven the identity.