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