Prove that
2sinxsiny = cos(x-y) - cos(x+y)
Prove that
sin2x + sin2y = 2sin(x+y)cos(x-y)
On the first, you have formulas for cos(a+-b). Use them on the right, and it will simplifiy quickly.
On the second, I wouldstart with the double angle formulas...on the left.
To prove the identities:
1. Prove: 2sin(x)sin(y) = cos(x-y) - cos(x+y)
Start with the formula for the cosine of the difference of two angles:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
Using the formulas for sine of the sum and difference of two angles (sin(A + B) and sin(A - B)), we can rewrite the above formula as:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
Now, set A = x and B = y:
cos(x - y) = cos(x)cos(y) + sin(x)sin(y)
Rearrange the equation:
cos(x - y) - cos(x)cos(y) = sin(x)sin(y)
Finally, dividing both sides by 2 gives:
(cos(x - y) - cos(x)cos(y))/2 = sin(x)sin(y)
Comparing the right side of this equation with 2sin(x)sin(y), we can see that they are equal. Therefore, the identity is proved.
2. Prove: sin(2x) + sin(2y) = 2sin(x + y)cos(x - y)
Start with the formula for the double angle of sine:
sin(2A) = 2sin(A)cos(A)
Let A = x + y, then:
sin(2(x + y)) = 2sin(x + y)cos(x + y)
Using the formula for sine of the sum of two angles, we can rewrite the equation as:
sin(2x + 2y) = 2sin(x + y)cos(x + y)
Now, using the formula for the sum of angles, we can rewrite the equation as:
sin(2x)cos(2y) + cos(2x)sin(2y) = 2sin(x + y)cos(x + y)
Remembering that cos(2A) = cos²(A) - sin²(A), we can rewrite the equation as:
sin(2x)(1 - 2sin²(y)) + (1 - 2sin²(x))sin(2y) = 2sin(x + y)cos(x + y)
Expanding the terms:
sin(2x) - 2sin²(x)sin²(y) + sin(2y) - 2sin²(x)sin²(y) = 2sin(x + y)cos(x + y)
Combining like terms:
sin(2x) + sin(2y) - 2sin²(x)sin²(y) - 2sin²(x)sin²(y) = 2sin(x + y)cos(x + y)
Simplifying the right side:
sin(2x) + sin(2y) - 4sin²(x)sin²(y) = 2sin(x + y)cos(x + y)
Using the formula for double angle of sine again, we get:
2sin(x)cos(x) + 2sin(y)cos(y) - 4sin²(x)sin²(y) = 2sin(x + y)cos(x + y)
Simplifying the left side:
2[sin(x)cos(x) + sin(y)cos(y) - 2sin²(x)sin²(y)] = 2sin(x + y)cos(x + y)
Now, we can see that the left and right sides of the equation are equal. Therefore, the identity is proved.