Question

The expression $\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}$ can be written as

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

Answer 1

Answer: (A)

$\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}$
$=\frac{{\sin }^{2}A}{\cos A\left(\sin A-\cos A\right)}+\frac{{\cos }^{2}A}{\sin A\left(\cos A-\sin A\right)}$
$=\frac{{\sin }^{3}A-{\cos }^{3}A}{\left(\sin A-\cos A\right)\cos A\sin A}$
$=\frac{\left(\sin A-\cos A\right)\left({\sin }^{2}A+\sin A\cos A+{\cos }^{2}A\right)}{\left(\sin A-\cos A\right)\sin A\cos A}$
$=\frac{1+\sin A\cos A}{\sin A\cos A}=1+\sec AcosecA$