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