The value of frac{sin^2 3A}{sin^2 A} - frac{c | vedclass

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Question

(A) Given expression: $E = \frac{\sin^2 3A}{\sin^2 A} - \frac{\cos^2 3A}{\cos^2 A}$$E = \frac{\sin^2 3A \cos^2 A - \cos^2 3A \sin^2 A}{\sin^2 A \cos^2

Vedclass · Accepted Answer

$8 \cos 2A$

Vedclass · Answer

(A) Given expression: $E = \frac{\sin^2 3A}{\sin^2 A} - \frac{\cos^2 3A}{\cos^2 A}$$E = \frac{\sin^2 3A \cos^2 A - \cos^2 3A \sin^2 A}{\sin^2 A \cos^2 A}$Using $\sin^2 x \cos^2 y - \cos^2 x \sin^2 y = \sin(x-y)\sin(x+y)$:$E = \frac{\sin(3A-A)\sin(3A+A)}{\sin^2 A \cos^2 A} = \frac{\sin 2A \sin 4A}{\sin^2 A \cos^2 A}$Using $\sin 2A = 2 \sin A \cos A$ and $\sin 4A = 2 \sin 2A \cos 2A = 4 \sin A \cos A \cos 2A$:$E = \frac{(2 \sin A \cos A)(4 \sin A \cos A \cos 2A)}{\sin^2 A \cos^2 A}$$E = \frac{8 \sin^2 A \cos^2 A \cos 2A}{\sin^2 A \cos^2 A} = 8 \cos 2A$

The value of $\frac{\sin^2 3A}{\sin^2 A} - \frac{\cos^2 3A}{\cos^2 A}$ is

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