Question

Verify the identity Cotx/tanx+cotx = 1-sin^2x

Answer 1

`\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{(cos(x))/(sin(x))}{(sin(x))/(cos(x))+(cos(x))/(sin(x))}\implies \cfrac{(cos(x))/(sin(x))}{(sin^2(x)+cos^2(x))/(cos(x)sin(x))}`


`\bf \cfrac{(cos(x))/(sin(x))}{(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)`