The expression tan A 1 - cot A + cot A 1 - tan A can be written as: [2013]

Question

The expression $\frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A}$ can be written as: [2013]

Smart Achievers · Accepted Answer

(2)

Given expression can be written as

$\frac{\sin A}{\cos A} \times \frac{\sin A}{\sin A - \cos A} + \frac{\cos A}{\sin A} \times \frac{\cos A}{\cos A - \sin A}$

$= \frac{1}{\sin A - \cos A} {\frac{\sin^{3} A - \cos^{3} A}{\cos A \sin A}}$

$∵$ $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$

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

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