What is the value of $$(1 + \cot A + \tan A) (\sin A - \cos A) \left(\frac{\sin A \cos A}{\sin^3 A - \cos^3 A}\right)$$​​

Correct option is A

Given:

​$$(1 + \cot A + \tan A) (\sin A - \cos A) \left(\frac{\sin A \cos A}{\sin^3 A - \cos^3 A}\right)$$​

Solution:

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