How do you prove that $sin 5 x = sin x ({cos}^{2} 2 x - {sin}^{2} 2 x) + 2 cos x cos 2 x sin 2 x$ $sin5x=sinx(cos^2 2x-sin^2 2x)+2cosxcos2xsin2x$ ?

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How do you prove that $sin 5 x = sin x ({cos}^{2} 2 x - {sin}^{2} 2 x) + 2 cos x cos 2 x sin 2 x$ $sin5x=sinx(cos^2 2x-sin^2 2x)+2cosxcos2xsin2x$ ?

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Answer 1 · 2016-10-22T22:28:18

Using the angle addition identity:

$sin (α + β) = sin (α) cos (β) + cos (α) sin (β)$ $sin(alpha + beta) = sin(alpha)cos(beta) + cos(alpha)sin(beta)$

along with the double angle identities:

$sin (2 θ) = 2 sin (θ) cos (θ)$ $sin(2theta) = 2sin(theta)cos(theta)$
$cos (2 θ) = {cos}^{2} (θ) - {sin}^{2} (θ)$ $cos(2theta) = cos^2(theta)-sin^2(theta)$

we have

$sin (5 x) = sin (x + 4 x)$ $sin(5x) = sin(x + 4x)$

$= sin (x) cos (4 x) + cos (x) sin (4 x)$ $=sin(x)cos(4x) + cos(x)sin(4x)$

$= sin (x) cos (2 \cdot 2 x) + cos (x) sin (2 \cdot 2 x)$ $=sin(x)cos(2*2x) + cos(x)sin(2*2x)$

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

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

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How do you prove that $sin 5 x = sin x ({cos}^{2} 2 x - {sin}^{2} 2 x) + 2 cos x cos 2 x sin 2 x$ $sin5x=sinx(cos^2 2x-sin^2 2x)+2cosxcos2xsin2x$ ?

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How do you prove that $sin 5 x = sin x ({cos}^{2} 2 x - {sin}^{2} 2 x) + 2 cos x cos 2 x sin 2 x$ $sin5x=sinx(cos^2 2x-sin^2 2x)+2cosxcos2xsin2x$ ?