Question

prove that cosA / CosA -SINA -COSA /COSA+SINA = TAN2A

Answer 1

Answer:

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Answer 2

The proof is given below:

To Prove

$\frac{\cos A}{\cos A-\sin A}-\frac{\cos A}cos A+\sin A}=\tan 2A$

LHS

$=\frac{\cos A}{\cos A-\sin A}-\frac{\cos A}cos A+\sin A}$

$=\cos A(\frac{1}{\cos A-\sin A}-\frac{1}{\cos A+\sin A}$

$=\cos A(\frac{(\cos A+\sin A)-(\cos A-\sin A)}{(\cos A-\sin A)(\cos A+\sin A)})$

$=\cos A(\frac{\cos A+\sin A-\cos A+\sin A}{\cos^A-\sin^A}$      [∵ a²-b²=(a+b)(a-b)]

$=\frac{\cos A\times 2\sin A}{\cos 2A}$     (∵$\cos^2 A-\sin^2 A=\cos 2A$)

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

$=\frac{\sin 2A}{\cos 2A}$           (∵ $\sin 2A=2\sin A\cos A$)

$=\tan 2A$

= RHS                          (Proved)

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