Question

Prove that: $\frac{cosA}{1-tanA}+\frac{sinA}{1-cotA}=sinA+cosA$

Answer 1

given equation is $\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}$

it can be written as

$\frac{\cos A}{1-\frac{\sin A}{\cos A}}+\frac{\sin A}{1-\frac{\cos A}{\sin A}}=\frac{{\cos }^{2}A}{\cos A-\sin A}+\frac{{\sin }^{2}A}{\sin A-\cos A}=\frac{{\cos }^{2}A-{\sin }^{2}A}{\cos A-\sin A}$

it is in the form of $a^2-b^2/a-b=a+b$

therefore it is equal to $cos\left(A\right)+sin\left(A\right)$