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(a) Evaluate : $\frac{2}{3}(\cos^4 30^\circ - \sin^4 45^\circ) - 3(\sin^2 60^\circ - \sec^2 45^\circ) + \frac{1}{4}\cot^2 30^\circ$.
OR
(b) Prove that $\frac{\sin A + \cos A}{\sin A - \cos A} + \frac{\sin A - \cos A}{\sin A + \cos A} = \frac{2}{1 - 2\cos^2 A}$.
OR
(b) Prove that $\frac{\sin A + \cos A}{\sin A - \cos A} + \frac{\sin A - \cos A}{\sin A + \cos A} = \frac{2}{1 - 2\cos^2 A}$.
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(a) $\frac{2}{3} [(\frac{\sqrt{3}}{2})^4 - (\frac{1}{\sqrt{2}})^4] - 3 [(\frac{\sqrt{3}}{2})^2 - (\sqrt{2})^2] + \frac{1}{4}(\sqrt{3})^2 = \frac{113}{24}$
OR
(b) $LHS = \frac{(\sin A + \cos A)^2 + (\sin A - \cos A)^2}{\sin^2 A - \cos^2 A} = \frac{2(\sin^2 A + \cos^2 A)}{1 - \cos^2 A - \cos^2 A} = \frac{2}{1 - 2\cos^2 A} = RHS$
OR
(b) $LHS = \frac{(\sin A + \cos A)^2 + (\sin A - \cos A)^2}{\sin^2 A - \cos^2 A} = \frac{2(\sin^2 A + \cos^2 A)}{1 - \cos^2 A - \cos^2 A} = \frac{2}{1 - 2\cos^2 A} = RHS$