Answer
Verify the following identity:
$(sec(u) - 1)/(sec(u) + 1) = (tan(u) - sin(u))/(tan(u) + sin(u))$
Work Step by Step
Verify the following identity:
$(sec(u) - 1)/(sec(u) + 1) = (tan(u) - sin(u))/(tan(u) + sin(u))$
Multiply numerator and denominator of $(sec(u) - 1)/(1 + sec(u))$ by $sin(u)$:
$(sin(u) (sec(u) - 1))/(sin(u) (1 + sec(u))) = ^?(tan(u) - sin(u))/(sin(u) + tan(u))$
$(sec(u) - 1) sin(u) = sec(u) sin(u) - sin(u) = (1/(cos(u))) sin(u) - sin(u) = tan(u) - sin(u)$:
$tan(u) - sin(u)/(sin(u) (1 + sec(u))) = ^?(tan(u) - sin(u))/(sin(u) + tan(u))$
$(1 + sec(u)) sin(u) = sin(u) + sec(u) sin(u) = sin(u) + (1/(cos(u))) sin(u) = sin(u) + tan(u)$:
$(tan(u) - sin(u))/sin(u) + tan(u) = ^?(tan(u) - sin(u))/(sin(u) + tan(u))$
The left hand side and right hand side are identical, thus verifying the identity.
