Prove that sin x - sin 3xcos x - cos 3x - tan 2x

Sin x + sin 3xcos x + cos 3x = tan 2x L.H.S= sin x + sin 3xcos x + cos 3x cos A + cos B = 2 cos(A+B2) cos(A B2) nbsp; sin A + sin B = 2 sin(A+B2) cos(A B2) ...

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Standard XII

Mathematics

Circular Measurement of Angle

Prove that si...

Question

Prove that $\frac{sin x + sin 3 x}{cos x + cos 3 x} = tan 2 x$

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Solution

$\frac{sin x + sin 3 x}{cos x + cos 3 x} = tan 2 x$
$L . H . S = \frac{sin x + sin 3 x}{cos x + cos 3 x}$
$cos A + cos B = 2 cos (\frac{A + B}{2}) cos (\frac{A - B}{2})$
$sin A + sin B = 2 sin (\frac{A + B}{2}) cos (\frac{A - B}{2})$
$= \frac{2 sin (\frac{x + 3 x}{2}) cos (\frac{x - 3 x}{2})}{2 cos (\frac{x + 3 x}{2}) cos (\frac{x - 3 x}{2})}$
$= \frac{sin (\frac{x + 3 x}{2})}{cos (\frac{x + 3 x}{2})}$
$= tan 2 x = R . H . S$
Hence proved.

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Circular Measurement of Angle

Standard XII Mathematics

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Prove that sinx+sin3xcosx+cos3x=tan2x

sinx+sin3xcosx+cos3x=tan2x L.H.S=sinx+sin3xcosx+cos3x cosA+cosB=2cos(A+B2)cos(A−B2) sinA+sinB=2sin(A+B2)cos(A−B2) =2sin(x+3x2)cos(x−3x2)2cos(x+3x2)cos(x−3x2) =sin(x+3x2)cos(x+3x2) =tan2x=R.H.S Hence proved.

Prove that $\frac{sin x + sin 3 x}{cos x + cos 3 x} = tan 2 x$