Verify the following identity:
(sec(x)-tan(x))^2 = (1-sin(x))/(1+sin(x))
Multiply both sides by 1+sin(x):
(1+sin(x)) (sec(x)-tan(x))^2 = ^?1-sin(x)
Write secant as 1/cosine and tangent as sine/cosine:
(1+sin(x)) (1/(cos(x))-(sin(x))/(cos(x)))^2 = ^?1-sin(x)
Put 1/(cos(x))-(sin(x))/(cos(x)) over the common denominator cos(x): 1/(cos(x))-(sin(x))/(cos(x)) = (1-sin(x))/(cos(x)):
(1-sin(x))/(cos(x))^2 (1+sin(x)) = ^?1-sin(x)
Multiply each exponent in (1-sin(x))/(cos(x)) by 2:
((1-sin(x))^2)/(cos(x)^2) (1+sin(x)) = ^?1-sin(x)
Multiply both sides by cos(x)^2:
(1-sin(x))^2 (1+sin(x)) = ^?cos(x)^2 (1-sin(x))
(1-sin(x)) cos(x)^2 = cos(x)^2-cos(x)^2 sin(x):
(1-sin(x))^2 (1+sin(x)) = ^?cos(x)^2-cos(x)^2 sin(x)
Divide both sides by sin(x)-1:
(sin(x)-1) (1+sin(x)) = ^?-cos(x)^2
(sin(x)-1) (1+sin(x)) = sin(x)^2-1:
sin(x)^2-1 = ^?-cos(x)^2
sin(x)^2 = 1-cos(x)^2:
1-cos(x)^2-1 = ^?-cos(x)^2
-1+1-cos(x)^2 = -cos(x)^2:
-cos(x)^2 = ^?-cos(x)^2
The left hand side and right hand side are identical:
Answer: | (identity has been verified)
