Question

Simplify.

sinx csc²x-sinx cot²x
Use algebra and the fundamental trigonometric identities.
Your answer should be a number or use a single trigonometric function.

Answer 1

`sin^2(\theta)=1-cos^2(\theta)\hspace{5em} csc(\theta)=\cfrac{1}{sin(\theta)} \hspace{5em} cot(\theta)=\cfrac{cos(\theta)}{sin(\theta)} \\\\[-0.35em] ~\dotfill\\\\ sin(x)csc^2(x)-sin(x)cot^2(x)\implies sin(x)\left[ \cfrac{1}{sin(x)} \right]^2-sin(x)\left[ \cfrac{cos(x)}{sin(x)} \right]^2 \\\\\\ sin(x)\cdot \cfrac{1}{sin^2(x)}-sin(x)\cdot \cfrac{cos^2(x)}{sin^2(x)}\implies \cfrac{1}{sin(x)}-\cfrac{cos^2(x)}{sin(x)} \\\\\\ \cfrac{1-cos^2(x)}{sin(x)}\implies \cfrac{sin^2(x)}{sin(x)}\implies sin(x)`