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Answer 1 · 2016-10-21T10:08:51

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$sin (3 x) = 3 sin x - 4 {sin}^{3} x$ $sin (3x)=3sin x-4sin^3 x$

Use the formulas
$sin (A + B) = sin A cos B + cos A sin B$ $sin (A+B) = sin A cos B + cos A sin B$

$sin 2 A = 2 sin A cos A$ $sin 2A = 2sin A cos A$

$cos 2 A = 1 - 2 {sin}^{2} A$ $cos 2A = 1- 2 sin ^ 2 A$

Left Side: $sin 3 x = sin (2 x + x) = sin 2 x cos x + cos 2 x sin x$ $sin 3x=sin(2x+x)=sin2xcosx+cos2xsin x$

$= 2 sin x cos x cos x + (1 - 2 {sin}^{2} x) sin x$ $=2sinx cosx cosx + (1-2sin^2 x)sin x$

$= 2 sin x {cos}^{2} x + sin x - 2 {sin}^{3} x$ $=2sinx cos^2 x+ sinx-2sin^3 x$

$= 2 sin x (1 - {sin}^{2} x) + sin x - 2 {sin}^{3} x$ $=2sinx(1-sin^2 x) + sinx-2 sin^3 x$

$= 2 sin x - 2 {sin}^{3} x + sin x - 2 {sin}^{3} x$ $=2 sin x-2sin^3 x + sin x -2 sin^3 x$

$= 3 sin x - 4 {sin}^{3} x$ $=3 sin x -4 sin^ 3 x$

$=$ $=$ Right Side