Prove that: ^2x- ^2x=4 2x cosec 2x - Vidyadip

Question

Prove that:
$\cot^2\text{x}-\tan^2\text{x}=4\cot2\text{x}\ \text{cosec}\ 2\text{x}$

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Answer

$\text{LHS}=\cot^2\text{x}=\tan^2\text{x}$
$=\frac{\cos^2\text{x}}{\sin^2\text{x}}-\frac{\sin^2\text{x}}{\cos^2\text{x}}$
$=\frac{(\cos^2\text{x})^2-(\sin^2\text{x})^2}{\sin^2\text{x}\cos^2\text{x}}$
$=\frac{(\cos^2\text{x}+\sin^2\text{x})(\cos^2\text{x}-\sin^2\text{x})}{(\sin\text{x}\cos\text{x}^2)}$ $[\because\text{a}^2-\text{b}^2-=(\text{a+b})(\text{a-b})]$
$=\frac{\cos2\text{x}}{\frac{1}{4}(2\sin\text{x}\cos\text{x})^2}$ $[\because\cos2\text{x}=\cos^2\text{x}-\sin^2\text{x}]$
$=\frac{4\cos2\text{x}}{\sin^22\text{x}}$
$=\frac{4\cos2\text{x}}{\sin^2\text{x}}.\frac{1}{\sin2\text{x}}$ $\Big[\because\text{cosec}\ \theta=\frac{1}{\sin\theta}\Big]$
$=4\cot1\text{x}.\text{cosec}2\text{x}=\text{RHS}$

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