Prove this identity using the product-to-sum identity for sine:
sin^2 x=1-cos(2x)/2
YOU MEAN
sin^2 x=[ 1-cos(2x) ] /2
or
2 sin^2 x = 1 - cos(2x)
cos 2 x = cos^2 x - sin^2 x
so
2 sin^2 x = 1 - cos^2 x + sin^2 x
sin^2 x = 1 - cos^2 x
or
sin^2 x + cos^2 x = 1
which is true
To prove the identity sin^2(x) = (1 - cos(2x))/2 using the product-to-sum identity for sine, we need to express the right side of the equation using that identity.
The product-to-sum identity for sine states that sin(A) * sin(B) = 1/2 * [cos(A - B) - cos(A + B)].
We can start by rewriting cos(2x) as cos(x + x). Applying the product-to-sum identity, we have:
cos(2x) = cos(x + x) = cos(x) * cos(x) - sin(x) * sin(x)
Now, let's substitute this expression back into the equation and simplify:
(1 - cos(2x))/2 = (1 - (cos(x) * cos(x) - sin(x) * sin(x)))/2
= (1 - cos^2(x) + sin^2(x))/2
= sin^2(x)
Therefore, we have proven that sin^2(x) = (1 - cos(2x))/2 using the product-to-sum identity for sine.