Question

Solve
$\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}=$?

• A. $\sec A\csc A+1$
• B. $\tan A+\cot A$
• C. $\sec A+\csc A$
• D. $\sin A\cos A+1$

Answer 1

The correct option is A $\sec A\csc A+1$

$\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}\Rightarrow \frac{\frac{\sin A}{\cos A}}{1-\frac{\cos A}{\sin A}}+\frac{\frac{\cos A}{\sin A}}{1-\frac{\sin A}{\cos A}}$

$\Rightarrow \frac{{\sin }^{2}A}{\cos A(\sin A-\cos A)}\frac{{\cos }^{2}A}{sinA(\sin A-\cos A)}$

$\Rightarrow \frac{{\sin }^{3}A-{\cos }^{3}A}{\sin A\cos A(\sin A-\cos A)}\frac{(1+\sin A\cos A)}{\sin A\cos A}$

$\Rightarrow 1+\sec A\csc A$