Question

Prove that : \(\sqrt{\frac{1 + \sin A}{1 - \sin A}} = \sec A + \tan A\).

Hint:

For expressions like \(\sqrt{\frac{1 \pm \sin A}{1 \mp \sin A}}\) or \(\sqrt{\frac{1 \pm \cos A}{1 \mp \cos A}}\), rationalizing the denominator is almost always the correct first step.

Answer 1

Step 1: Start with LHS
LHS = √[(1 + sin A) / (1 − sin A)]

Step 2: Rationalize the Denominator
Multiply numerator and denominator inside the root by (1 + sin A):

= √[(1 + sin A)(1 + sin A) / (1 − sin A)(1 + sin A)]

= √[(1 + sin A)² / (1 − sin²A)]

Step 3: Use Identity
We know:
1 − sin²A = cos²A

So expression becomes:
= √[(1 + sin A)² / cos²A]

Taking square root:
= (1 + sin A) / cos A

Step 4: Split the Fraction
= 1/cos A + sin A/cos A

= sec A + tan A

Final Answer:
LHS = RHS
Hence proved.