Answer
Verify the following identity:
$(1 + cos(x) + sin(x))/(1 + cos(x) - sin(x)) = (1 + sin(x))/(cos(x))$
Work Step by Step
Verify the following identity:
$(1 + cos(x) + sin(x))/(1 + cos(x) - sin(x)) = (1 + sin(x))/(cos(x))$
Cross multiply:
$cos(x) (1 + cos(x) + sin(x)) = ^?(1 + cos(x) - sin(x)) (1 + sin(x))$
$cos(x) (1 + cos(x) + sin(x)) = cos(x) + cos(x)^2 + cos(x) sin(x)$:
$cos(x) + cos(x)^2 + cos(x) sin(x) = ^?(1 + cos(x) - sin(x)) (1 + sin(x))$
$(1 + cos(x) - sin(x)) (1 + sin(x)) = 1 + cos(x) + cos(x) sin(x) - sin(x)^2$:
$cos(x) + cos(x)^2 + cos(x) sin(x) = ^?1 + cos(x) + cos(x) sin(x) - sin(x)^2$
$sin(x)^2 = 1 - cos(x)^2$:
$cos(x) + cos(x)^2 + cos(x) sin(x) = ^?1 + cos(x) + cos(x) sin(x) - 1 - cos(x)^2$
$-(1 - cos(x)^2) = cos(x)^2 - 1$:
$cos(x) + cos(x)^2 + cos(x) sin(x) = ^?1 + cos(x) + cos(x) sin(x) + cos(x)^2 - 1$
$1 + cos(x) + cos(x) sin(x) - 1 + cos(x)^2 = cos(x) + cos(x)^2 + cos(x) sin(x)$:
$cos(x) + cos(x)^2 + cos(x) sin(x) = ^?cos(x) + cos(x)^2 + cos(x) sin(x)$
The left hand side and right hand side are identical, thus verifying the identity.
