Question #238093

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)


1
Expert's answer
2021-09-21T11:29:29-0400

Solution:

(1) We know that

cos2A=12sin2A2sin2A=1cos2A ...(i)sinA=1cos2A2\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)=sinxf(x)=\sin x

Now, f(x+y)f(x)=sin(x+y)sinxf(x+y) – f(x)=\sin(x+y)-\sin x

=sinxcosy+cosxsinysinx=cosxsinysinx(1cosy)=cosxsinysinx(2sin2y2) [using (i)]=cosxsiny2sinxsin2y2=\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


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