Question
Prove.
$\tan A-\cot A=\frac{1-2 \cos ^2 A}{\sin A \cos A}$
$\tan A-\cot A=\frac{1-2 \cos ^2 A}{\sin A \cos A}$
✓
Answer
$\tan A-\cot A=\frac{\sin A}{\cos A}-\frac{\cos A}{\sin A}$
$=\frac{\sin ^2 A-\cos ^2 A}{\sin A \cos A}$
$=\frac{1-\cos ^2 A-\cos ^2 A}{\sin A \cos A}\left(\because \sin ^2 A =1-\cos ^2 A \right) $
$ =\frac{1-2 \cos 2}{\sin A \cos A}$
$=\frac{\sin ^2 A-\cos ^2 A}{\sin A \cos A}$
$=\frac{1-\cos ^2 A-\cos ^2 A}{\sin A \cos A}\left(\because \sin ^2 A =1-\cos ^2 A \right) $
$ =\frac{1-2 \cos 2}{\sin A \cos A}$
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