Answer
Verify the following identity:
$tan(u) (sin(u))/(tan(u) + sin(u)) = (tan(u) - sin(u))/(tan(u) sin(u))$
Work Step by Step
Verify the following identity:
$tan(u) (sin(u))/(tan(u) + sin(u)) = (tan(u) - sin(u))/(tan(u) sin(u))$
Cross multiply:
$sin(u)^2 tan(u)^2 = ^?(tan(u) - sin(u)) (sin(u) + tan(u))$
Write tangent as sine/cosine:
$sin(u)^2 ( (sin(u))/(cos(u)) ^2 ) = ^?((sin(u))/(cos(u)) - sin(u)) (sin(u) + (sin(u))/(cos(u)))$
$sin(u)^2 ((sin(u))/(cos(u)))^2 = (sin(u)^4)/(cos(u)^2)$:
$(sin(u)^4)/(cos(u)^2) = ^?(((sin(u))/(cos(u))) - sin(u)) (sin(u) + ((sin(u))/(cos(u))))$
Put $(sin(u))/(cos(u)) - sin(u)$ over the common denominator $cos(u)$: $(sin(u))/(cos(u)) - sin(u) = (sin(u) - cos(u) sin(u))/(cos(u))$:
$(sin(u)^4)/(cos(u)^2) = ^?(sin(u) - cos(u) sin(u))/(cos(u)) (sin(u) + (sin(u))/(cos(u)))$
Put $sin(u) + (sin(u))/(cos(u))$ over the common denominator $cos(u): sin(u) + (sin(u))/(cos(u)) = (sin(u) + cos(u) sin(u))/(cos(u))$:
$(sin(u)^4)/(cos(u)^2) = ^?(sin(u) - cos(u) sin(u))/(cos(u)) (sin(u) + cos(u) sin(u))/(cos(u))$
$((sin(u) - cos(u) sin(u)) (sin(u) + cos(u) sin(u)))/(cos(u) cos(u)) = ((sin(u) - cos(u) sin(u)) (sin(u) + cos(u) sin(u)))/cos(u)^2$:
$(sin(u)^4)/(cos(u)^2) = ^?((sin(u) - cos(u) sin(u)) (sin(u) + cos(u) sin(u)))/(cos(u)^2)$
Multiply both sides by $cos(u)^2$:
$sin(u)^4 = ^?(sin(u) - cos(u) sin(u)) (sin(u) + cos(u) sin(u))$
Divide both sides by $sin(u)^2$:
$sin(u)^2 = ^?-(cos(u) - 1) (1 + cos(u))$
$sin(u)^2 = 1 - cos(u)^2$:
$1 - cos(u)^2 = ^?-(cos(u) - 1) (1 + cos(u))$
$-(cos(u) - 1) (1 + cos(u)) = 1 - cos(u)^2$:
$1 - cos(u)^2 = ^?1 - cos(u)^2$
The left hand side and right hand side are identical, thus verifying the identity.
