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Prove the following:
(a) $(\cos \theta+\sin \theta)^2=1+\sin 2 \theta$
(b) $\frac{\cos 2 x}{\cos x-\sin x}=\cos x+\sin x$
(c) $\cos ^4 \alpha-\sin ^4 \alpha=\cos 2 \alpha$
(d) $\frac{1-\sin 2 x}{\sin x-\cos x}=\sin x-\cos x$
(e) $\frac{\sin \theta+\sin 2 \theta}{1+\cos \theta+\cos 2 \theta}=\tan \theta$
(f) $\frac{1+\cos 2 \mathrm{~A}}{\cos 2 \mathrm{~A}}=\frac{\tan 2 \mathrm{~A}}{\tan \mathrm{A}}$
(g) $\tan \mathrm{A}+\frac{\cos \mathrm{A}}{\sin \mathrm{A}}=\frac{2}{\sin 2 \mathrm{~A}}$
(h) $\left(\cos \frac{\theta}{2}+\sin \frac{\theta}{2}\right)^2=1+\sin \theta$
(i) $\frac{\sin 4 \theta-\sin 2 \theta}{\cos 4 \theta+\cos 2 \theta}=\tan \theta$
(j) $\sin 4 \theta=4 \sin \theta \cdot \cos \theta-8 \sin ^3 \theta \cdot \cos \theta$
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