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).

Question

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).​

Reiny · Accepted Answer

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

yuki · Answer

Reiny, how does cosy become cosy-1?

Jon · Answer

why is it 2sin^2 y/2 and not sin^2 y/2?

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).​

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).