if f(x)=sin x, show that f(x+y)-f(x)=cos x sin y-2sin x sin^2(y/2)
sin(x+y) = sinx cosy + cosx siny
so, use that to get
f(x+y)-f(x) = sinx cosy + cosx siny - sinx
= cosx siny + sinx (cosy - 1)
Now use the double-angle formula to finish it off.
To prove the equation f(x+y) - f(x) = cos(x)sin(y) - 2sin(x)sin^2(y/2), we need to use the trigonometric identities.
Let's start by expanding f(x+y) using the angle addition formula for sine:
f(x + y) = sin(x + y)
Applying the angle addition formula, we get:
= sin(x)cos(y) + cos(x)sin(y)
Now, we can rewrite f(x) using the given function f(x) = sin(x):
f(x) = sin(x)
Now, we can substitute these expressions back into the original equation:
f(x + y) - f(x) = (sin(x)cos(y) + cos(x)sin(y)) - sin(x)
Now, our goal is to manipulate the equation to match the right-hand side of the equation we want to prove.
First, let's combine like terms:
f(x + y) - f(x) = sin(x)cos(y) + cos(x)sin(y) - sin(x)
Next, we can separate terms and factor out sin(x) from the first two terms:
f(x + y) - f(x) = sin(x)(cos(y) + 1) + cos(x)sin(y)
Now, we can use the trigonometric identity cos^2(y/2) = 1/2 + (1/2)cos(y) to rewrite the first term:
cos(y) + 1 = 2cos^2(y/2)
Substituting this back into our equation:
f(x + y) - f(x) = sin(x)(2cos^2(y/2)) + cos(x)sin(y)
Next, we will use the identity sin^2(y/2) = (1 - cos(y))/2 to rewrite sin(y):
f(x + y) - f(x) = sin(x)(2cos^2(y/2)) + cos(x)(1 - cos(y))/2
Expanding the expression:
f(x + y) - f(x) = 2sin(x)cos^2(y/2) + (cos(x) - cos(x)cos(y))/2
Now, let's factor out 2sin(x)/2, and combine the two fractions:
f(x + y) - f(x) = sin(x)(2cos^2(y/2)) + cos(x)(1 - cos(y))/2
= sin(x)(2cos^2(y/2) + (1 - cos(y))/2)
Using the identity sin^2(y/2) = (1 - cos(y))/2:
f(x + y) - f(x) = sin(x)(2cos^2(y/2) + 2sin^2(y/2))
= sin(x)(2(cos^2(y/2) + sin^2(y/2)))
= sin(x)*2
Finally, the equation simplifies to:
f(x + y) - f(x) = 2sin(x)
As we can see, the final equation obtained does not match the right-hand side of the equation we want to prove, which is cos(x)sin(y) - 2sin(x)sin^2(y/2). Therefore, the given equation f(x+y) - f(x) = cos(x)sin(y) - 2sin(x)sin^2(y/2) is not correct.