Prove that 1-sin^2x/cosx = sin 2x/2sinx
To prove that
(1 - sin^2(x)) / cos(x) = sin(2x) / (2sin(x))
we will manipulate the left side of the equation and transform it into the right side.
Start with:
(1 - sin^2(x)) / cos(x)
Using the identity sin^2(x) + cos^2(x) = 1, we can rewrite sin^2(x) as (1 - cos^2(x)):
(1 - (1 - cos^2(x))) / cos(x)
= (1 - 1 + cos^2(x)) / cos(x)
= cos^2(x) / cos(x)
= cos(x)
Now let's look at the right side of the equation:
sin(2x) / (2sin(x))
Using the double angle formula for sine, sin(2x) = 2sin(x)cos(x):
(2sin(x)cos(x)) / (2sin(x))
= cos(x)
Both the left side and the right side of the equation simplify to cos(x), so we have proven that:
(1 - sin^2(x)) / cos(x) = sin(2x) / (2sin(x))