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(a) Evaluate : $2 \tan^2 45^\circ + \cos^2 30^\circ - \sin^2 90^\circ$.
OR
(b) Verify that $\cos 2A = \frac{1 - \tan^2 A}{1 + \tan^2 A}$ for $A = 30^\circ$.
OR
(b) Verify that $\cos 2A = \frac{1 - \tan^2 A}{1 + \tan^2 A}$ for $A = 30^\circ$.
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Solution: (a) $2(1)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 - (1)^2 = \frac{7}{4}$
OR
(b) $LHS = \cos 60^\circ = \frac{1}{2}$
$RHS = \frac{1 - \tan^2 30^\circ}{1 + \tan^2 30^\circ} = \frac{1 - \frac{1}{3}}{1 + \frac{1}{3}} = \frac{1}{2} = LHS$
OR
(b) $LHS = \cos 60^\circ = \frac{1}{2}$
$RHS = \frac{1 - \tan^2 30^\circ}{1 + \tan^2 30^\circ} = \frac{1 - \frac{1}{3}}{1 + \frac{1}{3}} = \frac{1}{2} = LHS$