verify the identity:
sin(x+y)-sin(x-y)=2cosx siny
hey stacey..
we know that,
sin(x+y)= sinx cosy + cosx siny
and
sin(x-y)= sinx cosy - cosx siny
so,
sin(x+y)-sin(x-y)
= {sinx cosy + cosx siny} - {sinx cosy - cosx siny}
= sinx cosy + cosx siny - sinx cosy + cosx siny
=cosx siny + cosx siny
=2cosx siny
hence verified.
To verify the given identity, we will start with the left-hand side of the equation and try to simplify it to match the right-hand side.
Starting with the left-hand side:
sin(x + y) - sin(x - y)
We can use the identities for sin(A + B) and sin(A - B) to expand this expression:
(sin x * cos y) + (cos x * sin y) - (sin x * cos y) + (cos x * sin y)
Now, we can see that the first and third terms cancel each other out:
(cos x * sin y) + (cos x * sin y)
Next, we can simplify the equation by factoring out cos x from both terms:
cos x * (sin y + sin y)
Now, we have:
2 * cos x * sin y
Comparing this to the right-hand side of the equation, which is 2 * cos x * sin y, we can see that they are equal.
Therefore, we have verified the given identity:
sin(x + y) - sin(x - y) = 2 * cos x * sin y
To verify the given identity sin(x+y) - sin(x-y) = 2cos(x) * sin(y), we will use trigonometric identities.
Let's start with the left-hand side (LHS) of the equation:
sin(x+y) - sin(x-y)
Using the trigonometric sum-to-product identity:
sin(A) - sin(B) = 2 * cos((A+B)/2) * sin((A-B)/2)
Applying this identity to LHS:
sin(x+y) - sin(x-y) = 2 * cos((x+y+x-y)/2) * sin((x+y-x+y)/2)
= 2 * cos(2x/2) * sin(2y/2)
= 2 * cos(x) * sin(y)
Thus, the left-hand side (LHS) is equal to 2cos(x) * sin(y), which matches the right-hand side (RHS) of the equation.
Therefore, the given identity sin(x+y) - sin(x-y) = 2cos(x) * sin(y) is verified.