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) \(\sin 4 \theta=4 \sin \theta \cdot \cos \theta-8 \sin ^3 \theta \cdot \cos \theta\)