For the functions f(x) = sin x, show with the aid of the elementary formula sin^2 A = 1/2(1-cos 2A) that f(x+y) - f(x) = cos x sin y-2 sin x sin^2 (1/2y).
f(x) = sinx
then ...
f(x+y) - f(x)
= sin(x+y) - sinx
= sinxcosy + cosxsiny - sinx
= sinx(cosy - 1) + cosxsiny , ---- we need the cosxsiny in our final result, ok so far
= - sinx(1 - cosy) + cosxsiny
aside: from sin^2 A = 1/2(1-cos 2A)
sin^2 (y/2) = 1/2(1 - cosy)
2sin^2 (y/2) = 1 - cosy <----- we have that in our last step above
so from
- sinx(1 - cosy) + cosxsiny
= -sinx(2sin^2 (y/2) + cosxsiny
= cosxsiny - 2sinx(sin^2 (y/2))
= RS
QED
Reiny, how does cosy become cosy-1?
why is it 2sin^2 y/2 and not sin^2 y/2?
To show that f(x+y) - f(x) is equal to cos x sin y - 2 sin x sin^2 (1/2y) using the formula sin^2 A = 1/2(1 - cos 2A), we'll start by evaluating f(x+y) and f(x) separately.
We have f(x) = sin x and f(x+y) = sin(x+y).
Using the sum identity for sine, we can express sin(x+y) as sin x cos y + cos x sin y.
Now, let's substitute these values into f(x+y) - f(x):
f(x+y) - f(x) = (sin x cos y + cos x sin y) - sin x
We can see that the term sin x on the right side cancels out with the -sin x on the left side, leaving us with:
f(x+y) - f(x) = sin x cos y + cos x sin y - sin x
Now, let's focus on simplifying sin x cos y - sin x:
sin x cos y - sin x can be factored as sin x (cos y - 1).
Since we know sin^2 A = 1/2(1 - cos 2A), we can rewrite sin x (cos y - 1) as 2 sin x sin^2 (1/2y).
Applying this simplification, we have:
f(x+y) - f(x) = sin x cos y + cos x sin y - sin x = 2 sin x sin^2 (1/2y)
Therefore, we have shown that f(x+y) - f(x) equals cos x sin y - 2 sin x sin^2 (1/2y) using the elementary formula sin^2 A = 1/2(1 - cos 2A).