Answer
True.
Work Step by Step
Verify the following identity:
$(sec(x))/(sec(x) - tan(x)) = sec(x) (sec(x) + tan(x))$
Multiply both sides by $sec(x) - tan(x)$:
$sec(x) = ^?sec(x) (sec(x) - tan(x)) (sec(x) + tan(x))$
Write secant as 1/cosine and tangent as sine/cosine:
$1/(cos(x)) = ^?1/(cos(x)) (1/(cos(x)) - (sin(x))/(cos(x))) (1/(cos(x)) + (sin(x))/(cos(x)))$
$(1/(cos(x))) ((1/(cos(x))) - ((sin(x))/(cos(x)))) ((1/(cos(x))) + ((sin(x))/(cos(x)))) = ((1/(cos(x)) - (sin(x))/(cos(x))) (1/(cos(x)) + (sin(x))/(cos(x))))/(cos(x))$:
$1/(cos(x)) = ^?((1/(cos(x)) - (sin(x))/(cos(x))) (1/(cos(x)) + (sin(x))/(cos(x))))/(cos(x))$
Put $1/(cos(x)) - (sin(x))/(cos(x))$ over the common denominator $cos(x)$:
$1/(cos(x)) - (sin(x))/(cos(x)) = (1 - sin(x))/(cos(x))$:
$1/(cos(x)) = ^?(1/(cos(x)) + (sin(x))/(cos(x)))/(cos(x)) (1 - sin(x))/(cos(x))$
Put $1/(cos(x)) + (sin(x))/(cos(x))$ over the common denominator $cos(x)$:
$1/(cos(x)) + (sin(x))/(cos(x)) = (1 + sin(x))/(cos(x))$:
$1/(cos(x)) = ^?(1 - sin(x))/(cos(x) cos(x)) (1 + sin(x))/(cos(x))$
$((1 - sin(x)) (1 + sin(x)))/(cos(x) cos(x) cos(x)) = ((1 - sin(x)) (1 + sin(x)))/cos(x)^3$:
$1/(cos(x)) = ^?((1 - sin(x)) (1 + sin(x)))/(cos(x)^3)$
Cross multiply:
$cos(x)^3 = ^?cos(x) (1 - sin(x)) (1 + sin(x))$
Divide both sides by $cos(x)$:
$cos(x)^2 = ^?-(sin(x) - 1) (1 + sin(x))$
$cos(x)^2 = 1 - sin(x)^2$:
$1 - sin(x)^2 = ^?-(sin(x) - 1) (1 + sin(x))$
$-(sin(x) - 1) (1 + sin(x)) = 1 - sin(x)^2$:
$1 - sin(x)^2 = ^?1 - sin(x)^2$
The left hand side and right hand side are identical, thus verifying the identity.
