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)
Solution:
(1) We know that
cos2A=1−2sin2A⇒2sin2A=1−cos2A ...(i)⇒sinA=1−cos2A2\cos2A=1-2\sin^2A \\ \Rightarrow 2\sin^2A=1-\cos2A \ ...(i) \\ \Rightarrow \sin A=\sqrt{\dfrac{{1-\cos 2A}}{2}}cos2A=1−2sin2A⇒2sin2A=1−cos2A ...(i)⇒sinA=21−cos2A
(2) Given f(x)=sinxf(x)=\sin xf(x)=sinx
Now, f(x+y)–f(x)=sin(x+y)−sinxf(x+y) – f(x)=\sin(x+y)-\sin xf(x+y)–f(x)=sin(x+y)−sinx
=sinxcosy+cosxsiny−sinx=cosxsiny−sinx(1−cosy)=cosxsiny−sinx(2sin2y2) [using (i)]=cosxsiny−2sinxsin2y2=\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=sinxcosy+cosxsiny−sinx=cosxsiny−sinx(1−cosy)=cosxsiny−sinx(2sin22y) [using (i)]=cosxsiny−2sinxsin22y
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