Prove the identity: $\sqrt{\frac{1 + \sin A}{1 - \sin A}} = \sec A + an A$. Show all steps clearly.

Q: Prove the identity: $\sqrt{\frac{1 + \sin A}{1 - \sin A}} = \sec A + \tan A$. Show all steps clearly.

To prove: $\sqrt{\frac{1 + \sin A}{1 - \sin A}} = \sec A + \tan A$ \nTake the Left Hand Side (LHS): $\text{LHS} = \sqrt{\frac{1 + \sin A}{1 - \sin A}}$ \nRationalize the denominator inside the square root by multiplying both numerator and denominator by $(1 + \sin A)$: $\text{LHS} = \sqrt{\frac{(1 + \sin A)(1 + \sin A)}{(1 - \sin A)(1 + \sin A)}}$ \nSimplify the numerator and denominator: $\text{LHS} = \sqrt{\frac{(1 + \sin A)^2}{1 - \sin^2 A}}$ \nUsing the fundamental trigonometric identity $\cos^2 A + \sin^2 A = 1$, we get $1 - \sin^2 A = \cos^2 A$: $\text{LHS} = \sqrt{\frac{(1 + \sin A)^2}{\cos^2 A}}$ \nTaking the square root of both numerator and denominator: $\text{LHS} = \frac{1 + \sin A}{\cos A}$ \nSplit the fraction into two separate terms: $\text{LHS} = \frac{1}{\cos A} + \frac{\sin A}{\cos A}$ \nUsing reciprocal and quotient definitions ($\frac{1}{\cos A} = \sec A$ and $\frac{\sin A}{\cos A} = \tan A$): $\text{LHS} = \sec A + \tan A$ \nSince $\text{LHS} = \text{RHS}$, the identity is proved.

Question

Prove the identity: $\sqrt{\frac{1 + \sin A}{1 - \sin A}} = \sec A + 	an A$. Show all steps clearly.

Accepted Answer

To prove: $\sqrt{\frac{1 + \sin A}{1 - \sin A}} = \sec A + 	an A$ 
Take the Left Hand Side (LHS): $	ext{LHS} = \sqrt{\frac{1 + \sin A}{1 - \sin A}}$ 
Rationalize the denominator inside the square root by multiplying both numerator and denominator by $(1 + \sin A)$: $	ext{LHS} = \sqrt{\frac{(1 + \sin A)(1 + \sin A)}{(1 - \sin A)(1 + \sin A)}}$ 
Simplify the numerator and denominator: $	ext{LHS} = \sqrt{\frac{(1 + \sin A)^2}{1 - \sin^2 A}}$ 
Using the fundamental trigonometric identity $\cos^2 A + \sin^2 A = 1$, we get $1 - \sin^2 A = \cos^2 A$: $	ext{LHS} = \sqrt{\frac{(1 + \sin A)^2}{\cos^2 A}}$ 
Taking the square root of both numerator and denominator: $	ext{LHS} = \frac{1 + \sin A}{\cos A}$ 
Split the fraction into two separate terms: $	ext{LHS} = \frac{1}{\cos A} + \frac{\sin A}{\cos A}$ 
Using reciprocal and quotient definitions ($\frac{1}{\cos A} = \sec A$ and $\frac{\sin A}{\cos A} = 	an A$): $	ext{LHS} = \sec A + 	an A$ 
Since $	ext{LHS} = 	ext{RHS}$, the identity is proved.

NCERT · Class 10 · Mathematics · Introduction to TrigonometryProve the identity: $\sqrt{\frac{1 + \sin A}{1 - \sin A}} = \sec A + \tan A$. Show all steps clearly.