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If $\sin 3A = 1$, find the value of $\cos 2A - \tan^2 45^\circ$.
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$3A = 90^\circ \Rightarrow A = 30^\circ$. $\cos 2A - \tan^2 45^\circ = \cos 60^\circ - \tan^2 45^\circ = \frac{1}{2} - 1 = -\frac{1}{2}$. ($\frac{1}{2} + 1\frac{1}{2}$ marks)