6. Integral of cos2x sin2x tan2x cos square x, sin square x

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\int \frac{{sin}^{2} x - {cos}^{2} x}{sin x cos x} d x

cos x + sin x = cos 2 x + sin 2 x

2 sin 2 x - cos 2 x = 1, x \neq (2 n + 1) \frac{π}{2}

sin 2 x + cos 2 x

\frac{1}{5}

sin 2 x + cos 2 x = \frac{1 + 2 tan x - {tan}^{2} x}{1 + {tan}^{2} x}

{cos}^{2} x + {sin}^{2} x + {tan}^{2} x

Prove that

(i) $\frac{sin 2 x}{1 - cos 2 x} = cot x$

(ii) $\frac{1 - cos2x}{1 + cos 2 x} = {tan}^{2} x$

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