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Question: If \(\sin A,\cos A\) and \(\tan A\) are in G.P., then \(\cos^{3}A + \cos^{2}A\) is equal to...

If sin⁡A,cos⁡A\sin A,\cos A and tan⁡A\tan A are in G.P., then cos⁡3A+cos⁡2A\cos^{3}A + \cos^{2}A is equal to

A

1

B

2

C

4

D

None of these

Answer

1

Explanation

Solution

We have sin⁡A,cos⁡A\sin A,\cos A and tan⁡A\tan A are in G.P.

cos⁡2A=sin⁡Atan⁡A=sin⁡2Acos⁡A⇒cos⁡3A−sin⁡2A=0\cos^{2}A = \sin A\tan A = \frac{\sin^{2}A}{\cos A} \Rightarrow \cos^{3}A - \sin^{2}A = 0

Hence cos⁡3A+cos⁡2A=sin⁡2A+cos⁡2A=1\cos^{3}A + \cos^{2}A = \sin^{2}A + \cos^{2}A = 1