Prove that ?? $\frac{sin x + sin 2 x + sin 3 x}{cos x + cos 2 x + cos 3 x} = tan 2 x$ $(Sinx+Sin2x+Sin3x)/(cosx+cos2x+cos3x) = tan2x$

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Prove that ?? $\frac{sin x + sin 2 x + sin 3 x}{cos x + cos 2 x + cos 3 x} = tan 2 x$ $(Sinx+Sin2x+Sin3x)/(cosx+cos2x+cos3x) = tan2x$

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Answer 1 · 2018-05-11T07:52:52

$L H S = \frac{sin x + sin 2 x + sin 3 x}{cos x + cos 2 x + cos 3 x}$ $LHS=(sinx+sin2x+sin3x)/(cosx+cos2x+cos3x)$

$= \frac{2 sin (\frac{3 x + x}{2}) \cdot cos (\frac{3 x - x}{2}) + sin 2 x}{2 cos (\frac{3 x + x}{2}) \cdot cos (\frac{3 x - x}{2}) + cos 2 x}$ $=(2sin((3x+x)/2)*cos((3x-x)/2)+sin2x)/(2cos((3x+x)/2)*cos((3x-x)/2)+cos2x$

$= \frac{2 sin 2 x \cdot cos x + sin 2 x}{2 cos 2 x \cdot cos x + cos 2 x}$ $=(2sin2x*cosx+sin2x)/(2cos2x*cosx+cos2x)$

$=(sin2xcancel((1+2cosx)))/(cos2xcancel((1+2cosx)))$

$=tan2x=RHS$

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Prove that ?? $\frac{sin x + sin 2 x + sin 3 x}{cos x + cos 2 x + cos 3 x} = tan 2 x$ $(Sinx+Sin2x+Sin3x)/(cosx+cos2x+cos3x) = tan2x$

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Prove that ??(Sinx+Sin2x+Sin3x)/(cosx+cos2x+cos3x) = tan2xsinx+sin2x+sin3xcosx+cos2x+cos3x=tan2x(Sinx+Sin2x+Sin3x)/(cosx+cos2x+cos3x) = tan2x

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Prove that ?? $\frac{sin x + sin 2 x + sin 3 x}{cos x + cos 2 x + cos 3 x} = tan 2 x$ $(Sinx+Sin2x+Sin3x)/(cosx+cos2x+cos3x) = tan2x$