Prove that:
2sinxsiny = cos(x-y) - cos(x+y)
To price this equation first subtract cos(x-y) and cos(x+y)
=cosXcosY + sinXsinY
-(cosXcosY-sinXsinY)
= 2sinXsinY
Hence , proved!
To prove the given equation:
2sin(x)sin(y) = cos(x-y) - cos(x+y)
We can start by simplifying the right side of the equation using the trigonometric identity for the difference of angles:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
Using this identity, we can rewrite the right side of the equation as:
cos(x - y) - cos(x + y) = cos(x)cos(-y) + sin(x)sin(-y) - cos(x)cos(y) - sin(x)sin(y)
Remember that cos(-y) = cos(y) and sin(-y) = -sin(y), so we can simplify the equation further:
= cos(x)cos(y) - sin(x)sin(y) - cos(x)cos(y) + sin(x)sin(y)
The terms in the middle cancel each other out:
= 0
Therefore, we have shown that:
2sin(x)sin(y) = 0
The equation is indeed true for all values of x and y.
Rewrite the right side, using
cos (x + y) = cos x cos y - sin x sin y
cos (x - y) = cos x cos y + sin x sin y
You should be familiar with those two identities.