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