verify the following identity:
sin(x+y)*sin(x-y)=sin^2x-sin^2y
To verify the given identity sin(x+y) * sin(x-y) = sin^2x - sin^2y, we can use trigonometric identities to simplify both sides individually and then compare them.
Starting with the left side of the equation:
sin(x+y) * sin(x-y)
We can expand this using the product-to-sum identity:
sin(x+y) = sin(x) * cos(y) + cos(x) * sin(y)
sin(x-y) = sin(x) * cos(y) - cos(x) * sin(y)
Now substituting these values back into the identity:
(sin(x) * cos(y) + cos(x) * sin(y)) * (sin(x) * cos(y) - cos(x) * sin(y))
Expanding this equation further by applying the distributive property:
(sin^2(x) * cos^2(y)) - cos^2(x) * sin^2(y)
Now, we can use the identity sin^2θ + cos^2θ = 1 to simplify further:
1 - cos^2(x) * sin^2(y)
Finally, substituting sin^2(x) = 1 - cos^2(x) (from the identity above):
1 - (1 - cos^2(x)) * sin^2(y)
Expanding and rearranging terms:
1 - sin^2(y) + cos^2(x) * sin^2(y)
Now, using the identity 1 - sin^2θ = cos^2θ:
cos^2(y) + cos^2(x) * sin^2(y)
This expression matches the right side of the given identity sin^2x - sin^2y.
Therefore, we have verified that sin(x+y) * sin(x-y) = sin^2x - sin^2y using trigonometric identities and simplifications.