Simplify: Sin2x/1-Cos2x
To simplify the expression (sin2x)/(1 - cos2x), we can use the trigonometric identity:
sin^2(x) + cos^2(x) = 1
First, let's write sin^2(2x) as (sin(2x))^2:
(sin(2x))^2 / (1 - cos(2x))
Now, we can use another trigonometric identity:
sin(2x) = 2sin(x)cos(x)
Replacing sin(2x) with 2sin(x)cos(x):
(2sin(x)cos(x))^2 / (1 - cos(2x))
Expanding (2sin(x)cos(x))^2:
4sin^2(x)cos^2(x) / (1 - cos(2x))
Now, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to replace sin^2(x) and cos^2(x):
4(1 - cos^2(x))(cos^2(x)) / (1 - cos(2x))
Multiplying out the numerator:
4cos^2(x) - 4cos^4(x) / (1 - cos(2x))
Now, we can distribute the negative sign in the numerator:
4cos^2(x) - 4cos^4(x) / (1 - cos(2x))
Finally, we have simplified the expression:
4cos^2(x) - 4cos^4(x) / (1 - cos(2x))