Answer
Verify the following identity:
$(sin(y)^3 - csc(y)^3)/(sin(y) - csc(y)) = sin(y)^2 + csc(y)^2 + 1$
Work Step by Step
Verify the following identity:
$(sin(y)^3 - csc(y)^3)/(sin(y) - csc(y)) = sin(y)^2 + csc(y)^2 + 1$
Cancel $csc(y) - sin(y)$ from the numerator and denominator.
$(sin(y)^3 - csc(y)^3)/(sin(y) - csc(y))$
$= ((csc(y) - sin(y)) (csc(y)^2 + csc(y) sin(y) + sin(y)^2))/(csc(y) - sin(y))$
$= csc(y)^2 + csc(y) sin(y) + sin(y)^2$:
$csc(y)^2 + csc(y) sin(y) + sin(y)^2 = ^?1 + csc(y)^2 + sin(y)^2$
Subtract $csc(y)^2 + sin(y)^2$ from both sides:
$csc(y) sin(y) = ^?1$
Write cosecant as 1/sine:
$1/(sin(y)) sin(y) = ^?1$
$(1/(sin(y))) sin(y) = 1$:
$1 = ^?1$
The left hand side and right hand side are identical, thus verifying the identity.
