146
Prove that $\sec A (1 – \sin A) (\sec A + \tan A) = 1$.
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Sol. LHS = $\sec A (1 – \sin A) (\sec A + \tan A)$
$= \frac{1}{\cos A} (1 – \sin A) (\frac{1}{\cos A} + \frac{\sin A}{\cos A})$
$= \frac{1}{\cos A} (1 – \sin A) (\frac{1 + \sin A}{\cos A})$
$= \frac{1 - \sin^2 A}{\cos^2 A} = \frac{\cos^2 A}{\cos^2 A} = 1=RHS$
$= \frac{1}{\cos A} (1 – \sin A) (\frac{1}{\cos A} + \frac{\sin A}{\cos A})$
$= \frac{1}{\cos A} (1 – \sin A) (\frac{1 + \sin A}{\cos A})$
$= \frac{1 - \sin^2 A}{\cos^2 A} = \frac{\cos^2 A}{\cos^2 A} = 1=RHS$