Quantitative Aptitude Ques 564

Question: If $ \tan A+\sin ,A=p $ and $ tan,A-\sin A=q, $ then

Options:

A) $ p^{2}+q^{2}=4\sqrt{pq} $

B) $ p^{2}-q^{2}=4\sqrt{pq} $

C) $ p^{2}-q^{2}=\sqrt{pq} $

D) $ p^{2}-q^{2}=2\sqrt{pq} $

Show Answer

Answer:

Correct Answer: B

Solution:

  • $ \tan A+\sin A=p, $ $ \tan A-\sin A=q $ $ {{(\tan A+\sin A)}^{2}}=p^{2}, $ $ {{(\tan A-\sin A)}^{2}}=q^{2} $ $ p^{2}-q^{2}=4\tan A\sin A=\frac{4{{\sin }^{2}}A}{\cos A} $ $ 4\sqrt{pq}=4\sqrt{({{\tan }^{2}}A-{{\sin }^{2}}A)} $ $ =4\sqrt{\frac{{{\sin }^{2}}A,(1-{{\cos }^{2}}A)}{{{\cos }^{2}}A}}=\frac{4{{\sin }^{2}}A}{\cos A} $

$ \therefore $ $ p^{2}-q^{2}=4\sqrt{pq} $

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