Question

Verify the identity.

2sec^2A=1/1-SinA+1/1+SinA

Answer 1

$\sf{\underline{To\:verify:}}$

$\boxed{2 \: {sec}^{2} A = \frac{1}{1 - sinA} + \frac{1}{1 + sinA}}$

The right side is more complicated, so we start with it.

$\sf{\underline{RHS:}}$


$\huge\sf = \frac{1}{1 - sinA} + \frac{1}{1 + sinA}$


$\huge\sf = \frac{ 1 + sinA + 1 - sinA}{(1 - sinA)(1 + sinA)}$


$\huge\sf = \frac{2}{1 - {sin}^{2}A }$


$\huge\sf = \frac{2}{ {cos}^{2}A }$


$\sf{\underline{By\:using\:identity:}}$ $\boxed{ {cos}^{2} A = 1 - {sin}^{2} A}$


$\huge\sf = 2 {sec}^{2} A$


$\sf{\underline{By\:using\:identity:}}$ $\boxed{{sec}^{2} A = \frac{1}{cos ^{2} A}}$


$\huge\sf = LHS$


$\sf{\underline{Hence\:verified:}}$

$\boxed{2 \: {sec}^{2} A = \frac{1}{1 - sinA} + \frac{1}{1 + sinA}}$