How do I verify this identity? (I've tried a lot but it's just not working)
(sin2x + sin2y) = 2sin(x+y)cos(x-y)
To verify the given identity, you can use trigonometric identities and properties to manipulate both sides of the equation until they are equal.
Here's a step-by-step guide on how to verify the identity:
Step 1: Expand the left side of the equation using the double angle formula for sine:
sin(2x) = 2sin(x)cos(x) ...Equation 1
sin(2y) = 2sin(y)cos(y) ...Equation 2
(sin2x + sin2y) = (2sin(x)cos(x) + 2sin(y)cos(y)) ...Expanding using Equations 1 and 2
Step 2: Apply the sum-to-product identity for sin to the right side of the equation:
sin(x+y) = sin(x)cos(y) + cos(x)sin(y) ...Sum-to-product identity for sin
sin(x-y) = sin(x)cos(y) - cos(x)sin(y) ...Sum-to-product identity for sin
2sin(x+y)cos(x-y) = 2(sin(x)cos(y) + cos(x)sin(y))(sin(x)cos(y) - cos(x)sin(y)) ...Expanding using the sum-to-product identities
Step 3: Simplify the right side by multiplying the binomials:
2(sin(x)cos(y) + cos(x)sin(y))(sin(x)cos(y) - cos(x)sin(y))
= 2((sin(x)cos(y))^2 - (cos(x)sin(y))^2) ...Difference of squares
Step 4: Apply the double angle formulas for sine and cosine:
(sin(x)cos(y))^2 - (cos(x)sin(y))^2
= (sin(x))^2(cos(y))^2 - (cos(x))^2(sin(y))^2 ...Using double angle formulas
= sin^2(x)cos^2(y) - cos^2(x)sin^2(y)
Step 5: Combine like terms:
sin^2(x)cos^2(y) - cos^2(x)sin^2(y)
= sin^2(x)(1 - sin^2(y)) - (1 - cos^2(x))sin^2(y) ...Using the Pythagorean identity: sin^2(x) + cos^2(x) = 1
= sin^2(x) - sin^2(x)sin^2(y) - sin^2(y) + cos^2(x)sin^2(y)
= sin^2(x) - sin^2(y) + sin^2(x)sin^2(y) + cos^2(x)sin^2(y)
Step 6: Factor out sin^2(x):
sin^2(x) - sin^2(y) + sin^2(x)sin^2(y) + cos^2(x)sin^2(y)
= sin^2(x)(1 - sin^2(y)) + cos^2(x)sin^2(y)
= sin^2(x)cos^2(y) + cos^2(x)sin^2(y)
Step 7: Substitute the result from Step 6 back into the original equation:
(sin2x + sin2y) = 2sin(x+y)cos(x-y)
sin^2(x)cos^2(y) + cos^2(x)sin^2(y) = 2sin(x+y)cos(x-y)
Step 8: Both sides of the equation are now equal, so the identity has been verified.
Remember to check your calculations and make sure you haven't made any mistakes along the way. It can be helpful to simplify both sides before comparing them to ensure that they are in the same form.