Prove.1 A+ A = A- A - Vidyadip

Question

Prove.$\frac{1}{\sec A+\tan A}=\sec A-\tan A$

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Answer

$ \text { LHS }=\frac{1}{\sec A +\tan A } $
$ =\frac{1}{\frac{1}{\cos A}+\frac{\sin A}{\cos A}} $
$=\frac{1}{\frac{1+\sin A }{\cos A }}$
$ =\frac{\cos A }{1+\sin A } \times \frac{1-\sin A }{1+\sin A } $
$=\frac{\cos A(1-\sin A)}{(1)^2-\sin 2 A}$
$ =\frac{\cos A(1-\sin A)}{\cos ^2 A} $
$ =\frac{1}{\cos A}-\frac{\sin A}{\cos A} $
$ =\sec A -\tan A $
LHS $=$ RHS
Hence proved.

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