Answer
Verify the following identity:
$(sin(a) - tan(a)) (cos(a) - cot(a)) = (cos(a) - 1) (sin(a) - 1)$
Work Step by Step
Verify the following identity:
$(sin(a) - tan(a)) (cos(a) - cot(a)) = (cos(a) - 1) (sin(a) - 1)$
$(cos(a) - cot(a)) (sin(a) - tan(a))$
$= cos(a) sin(a) - cot(a) sin(a) - cos(a) tan(a) + cot(a) tan(a)$:
$cos(a) sin(a) - cot(a) sin(a) - cos(a) tan(a) + cot(a) tan(a) = ^?(cos(a) - 1) (sin(a) - 1)$
$(cos(a) - 1) (sin(a) - 1) = 1 - cos(a) - sin(a) + cos(a) sin(a)$:
$cos(a) sin(a) - cot(a) sin(a) - cos(a) tan(a) + cot(a) tan(a) = ^?1 - cos(a) - sin(a) + cos(a) sin(a)$
Write cotangent as cosine/sine and tangent as sine/cosine:
$cos(a) sin(a) - (cos(a))/(sin(a)) sin(a) - (sin(a))/(cos(a)) cos(a) + (cos(a))/(sin(a)) (sin(a))/(cos(a))$
$= ^?1 - cos(a) - sin(a) + cos(a) sin(a)$
$cos(a) sin(a) - ((cos(a))/(sin(a))) sin(a) - cos(a) ((sin(a))/(cos(a))) + ((cos(a))/(sin(a))) ((sin(a))/(cos(a)))$
$= 1 - cos(a) - sin(a) + cos(a) sin(a)$:
$1 - cos(a) - sin(a) + cos(a) sin(a) = ^?1 - cos(a) - sin(a) + cos(a) sin(a)$
The left hand side and right hand side are identical, thus verifying the identity.
