Question

Prove that $\frac{\cot \alpha .\cos \alpha }{\cot \alpha +\cos \alpha }=\frac{\cot \alpha -\cos \alpha }{\cot \alpha .\cos \alpha }$

Answer 1

Cross multiplying given question, we get

$co{t}^2\alpha .co{s}^2\alpha =co{t}^2\alpha -co{s}^2\alpha$

Solving RHS, we get

$⟹co{t}^2\alpha -co{s}^2\alpha$

$⟹\frac{co{s}^2\alpha }{si{n}^2\alpha }-co{s}^2\alpha$

$⟹\frac{co{s}^2\alpha (1-si{n}^2\alpha )}{si{n}^2\alpha }$

$⟹\frac{co{s}^2\alpha \left(co{s}^2\alpha \right)}{si{n}^2\alpha }$

$⟹co{s}^2\alpha .co{t}^2\alpha$ = LHS,

Hence proved