Question
Prove the following identitie:
$(1 + \tan A + \sec A)(1 + \cot A - \cos ec A) = 2$
$(1 + \tan A + \sec A)(1 + \cot A - \cos ec A) = 2$
✓
Answer
$(1 + \tan A + \sec A) (1 + \cot A - \cos ec A)$
$=1 + \cot A - cosec A + \tan A + 1 - \sec A + \sec A + cosec A - cosec A \sec A$
$=2+\frac{\cos A}{\sin A}+\frac{\sin A}{\cos A}-\frac{1}{\sin A \cos A}$
$=2+\frac{\cos ^2 A+\sin ^2 A}{\sin A \cos A}-\frac{1}{\sin A \cos A}$
$=2+\frac{1}{\sin A \cos A}-\frac{1}{\sin A \cos A}$
$=2$
$=1 + \cot A - cosec A + \tan A + 1 - \sec A + \sec A + cosec A - cosec A \sec A$
$=2+\frac{\cos A}{\sin A}+\frac{\sin A}{\cos A}-\frac{1}{\sin A \cos A}$
$=2+\frac{\cos ^2 A+\sin ^2 A}{\sin A \cos A}-\frac{1}{\sin A \cos A}$
$=2+\frac{1}{\sin A \cos A}-\frac{1}{\sin A \cos A}$
$=2$
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