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Question

Question: \[\frac{\sin 3A - \cos\left( \frac{\pi}{2} - A \right)}{\cos A + \cos(\pi + 3A)} =\]...

sin⁡3A−cos⁡(π2−A)cos⁡A+cos⁡(π+3A)=\frac{\sin 3A - \cos\left( \frac{\pi}{2} - A \right)}{\cos A + \cos(\pi + 3A)} =

A

tan⁡A\tan A

B

cot⁡A\cot A

C

tan⁡2A\tan 2A

D

cot⁡2A\cot 2A

Answer

cot⁡2A\cot 2A

Explanation

Solution

sin⁡3A−cos⁡(π2−A)cos⁡A+cos⁡(π+3A)=sin⁡3A−sin⁡Acos⁡A−cos⁡3A\frac{\sin 3A - \cos\left( \frac{\pi}{2} - A \right)}{\cos A + \cos(\pi + 3A)} = \frac{\sin 3A - \sin A}{\cos A - \cos 3A}

=2cos⁡2Asin⁡A2sin⁡2Asin⁡A=cos⁡2Asin⁡2A=cot⁡2A\frac{2\cos 2A\sin A}{2\sin 2A\sin A} = \frac{\cos 2A}{\sin 2A} = \cot 2A.