Prove the following identities:

(a) [tex]\sin^2 A - \cos^2 A = 2 \sin^2 A - 1[/tex]

(b) [tex]\cos^2 \theta \left(1 + \tan^2 \theta\right) = 1[/tex]

(c) [tex]\cos \theta \left(1 + \tan^2 \theta\right) = \sec \theta[/tex]

(d) [tex]\sin^2 \theta - \cos^2 \theta = 1 - 2 \cos^2 \theta[/tex]

(e) [tex]\sin^2 \theta \left(1 + \cot^2 \theta\right) = 1[/tex]

(f) [tex]\sin A \left(1 + \cot^2 A\right) = \operatorname{cosec} A[/tex]

(g) [tex]\sqrt{1 - \sin^2 A} \times \frac{1}{\sin A} = \cot A[/tex]

(h) [tex]\cos A \sqrt{1 + \tan^2 A} + \sin A \sqrt{1 + \cot^2 A} = 2[/tex]

(i) [tex]\frac{1}{\tan \theta + \cot \theta} = \sin \theta \cdot \cos \theta[/tex]

(j) [tex]\frac{\sin^4 \theta - \cos^4 \theta}{\sin \theta + \cos \theta} = \sin \theta - \cos \theta[/tex]

(k) [tex]\sqrt{\frac{1 - \cos A}{1 + \cos A}} = \operatorname{cosec} A - \cot A[/tex]

(l) [tex]\sqrt{\frac{1 - \sin \alpha}{1 + \sin \alpha}} = \sec \alpha - \tan \alpha[/tex]

(m) [tex]\sqrt{\frac{1 + \sin \alpha}{1 - \sin \alpha}} = \sec \alpha + \tan \alpha[/tex]

(n) [tex]\frac{1 + \cos x}{1 - \cos x} = (\operatorname{cosec} x + \cot x)^2[/tex]