Question

$\frac{\sin 3A-\cos (\frac{\pi }{2}-A)}{\cos A+\cos (\pi +3A)}$

Answer 1

$\frac{\sin 3A-\cos (\frac{\pi }{2}-A)}{\cos A+\cos (\pi +3A)}$

This can be written as

$\Rightarrow \frac{\sin 3A-\sin A}{\cos A-\cos 3A}$

$\Rightarrow \frac{3\sin A-4{\sin }^{3}A-\sin A}{\cos A-4{\cos }^{3}A+3\cos A}$

$\Rightarrow \frac{2\sin A-4{\sin }^{3}A}{4\cos A-4{\cos }^{3}A}$

$\Rightarrow \frac{\sin A-2{\sin }^{3}A}{2\cos A-2{\cos }^{3}A}$

$\Rightarrow \frac{\sin A(1-2{\sin }^{2}A)}{2\cos A(1-{\cos }^{2}A)}$

$\Rightarrow \frac{\sin A\cos 2A}{2\cos A{\sin }^{2}A}$

$\Rightarrow \frac{\cos 2A}{2\cos A\sin A}$

$\Rightarrow \frac{\cos 2A}{\sin 2A}$

$\Rightarrow \cot 2A$