Question

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

• A. $\sec A \, cosec A + 1$
• B. $\tan A + \cot A$
• C. $\sec A + cosec A$
• 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 A cosecA$