Final answer:
By substituting the double-angle identity for cosine and simplifying using Pythagorean identities, we have verified that the trigonometric identity 1 - sin(x) = (1 + cos(2x)) / (2(1 - sin(x))) holds true.
Step-by-step explanation:
To verify the identity 1 - sin(x) = (1 + cos(2x)) / (2(1 - sin(x))), let's start by examining the right-hand side of the equation.
Recall the double-angle identity for cosine, which states: cos(2x) = 1 - 2sin2(x), or alternatively, cos(2x) = 2cos2(x) - 1. We can use the first form for our purposes.
Now, substitute the double-angle identity into the right-hand side:
(1 + cos(2x)) / (2(1 - sin(x))) = (1 + (1 - 2sin2(x))) / (2(1 - sin(x)))
Simplify the numerator:
(2 - 2sin2(x)) / (2(1 - sin(x)))
Factor out a 2 from the numerator:
2(1 - sin2(x)) / (2(1 - sin(x)))
Recognize that 1 - sin2(x) is a Pythagorean identity and is equal to cos2(x). So we can further simplify:
2cos2(x) / (2(1 - sin(x)))
Cancel out the factor of 2:
cos2(x) / (1 - sin(x))
Looking at the denominator, we can use another Pythagorean identity, cos2(x) = 1 - sin2(x), which matches our denominator:
The expression simplifies to 1 - sin(x), which is exactly the left-hand side of our original identity.
So we have verified that indeed 1 - sin(x) = (1 + cos(2x)) / (2(1 - sin(x))).