Answer to Question #238093 in Trigonometry for Ligaya

Question

For the function f(x) = sin x, show with the aid elementary formula

sinA=1/2(1–cos2A)

f(x+y) – f(x) = cos x sin y – 2sin x sin^2 (1/2y)

Answer 1

Solution:

(1) We know that

$\cos2A=1-2\sin^2A \\ \Rightarrow 2\sin^2A=1-\cos2A \ ...(i) \\ \Rightarrow \sin A=\sqrt{\dfrac{{1-\cos 2A}}{2}}$

(2) Given $f(x)=\sin x$

Now, $f(x+y) – f(x)=\sin(x+y)-\sin x$

$=\sin x \cos y+\cos x \sin y-\sin x \\=\cos x \sin y-\sin x (1-\cos y) \\=\cos x \sin y-\sin x (2\sin^2\frac y2) \ [using \ (i)] \\=\cos x \sin y-2\sin x \sin^2\frac y2$