Prove the following identity : secA(1 - sinA)(secA + tanA) = 1 - Vidyadip

Question

Prove the following identity :
secA(1 - sinA)(secA + tanA) = 1

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Answer

$
\begin{aligned}
& \text { LHS }=\sec A(1-\sin A)(\sec A+\tan A) \\
& =\frac{1}{\cos A}(1-\sin A)\left(\frac{1}{\cos A}+\frac{\sin A}{\cos A}\right) \\
& =\frac{(1-\sin A)}{\cos A}\left(\frac{1+\sin A}{\cos A}\right)=\left(\frac{1-\sin ^2 A}{\cos ^2 A}\right) \\
& =\left(\frac{\cos ^2 A}{\cos ^2 A}\right) \\
& =1=\text { RHS }
\end{aligned}
$

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