Question

Prove that:
$\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}=\sin A+\cos A$

Answer 1

LHS $=\frac{cosA}{1-tanA}+\frac{sinA}{1-cotA}\Rightarrow \frac{cosA}{1-\frac{sinA}{cosA}}+\frac{sinA}{1-\frac{cosA}{sinA}}$

$\Rightarrow \frac{co{s}^{2}A}{cosA-sinA}+\frac{si{n}^{2}A}{sinA-cosA}=\frac{co{s}^{2}A-si{n}^{2}A}{cosA-sinA}$

$\frac{(cosA+sinA)(cosA-sinA)}{(cosA-sinA)}\Rightarrow cosA+sinA=RHS$

Hence [LHS=RHS]