 Krishna
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Step 1: Take down the given equation

GIVEN: \sqrt{\frac{1 + \sin A}{1 - \sin A}} = \sec A + \tan A

Step 2: Take the L.H.S of the equation and think about the possible ways to prove R.H.S.

NOTE: \sqrt{\frac{1 + \sin A}{1 - \sin A}}

Step 3: Rationalize the irrational denominator of L.H.S

EXPLANATION; You can't solve an equation that contains a fraction with an

irrational denominator, which means that the denominator contains a term

with a radical sign. This includes square, cube and higher roots. Getting rid

of the radical sign is called rationalizing the denominator.

EXAMPLE: \sqrt{\frac{1 + \sin A}{1 - \sin A}}

Divide and multiply by 1 + \sin A

=   \sqrt{\frac{1 + \sin A}{1 - \sin A} \frac{1 + \sin A}{1 + \sin A}}

=   \sqrt{\frac{(1 + \sin A )^2}{(1 - \sin A)(1 + \sin A)}}

=   \sqrt{\frac{(1 + \sin A)^2}{1^2 - \sin^2 A}}

=   \frac{1 + \sin A}{\cos^2 A}          (Since, \sin^2 \theta + \cos^2 \theta = 1 )

Separate the denominator and simplify for R.H.S

=   \frac{1}{\cos A} + \frac{ \sin A}{\cos A}

= \sec A + \tan A

(since, \sec \theta = \frac{1}{\cos \theta}

\tan \theta = \frac{\sin \theta}{\cos \theta} )

Hence, proved