Prove that- sin 5x- sin 3xcos 5x - cos 3x - tan 4x

Sin 5x+ sin 3xcos 5x + cos 3x= tan 4x L.H.S = sin 5x + sin 3xcos 5x + cos 3x cos A + cos B = 2 cos( A+B2) cos( A B2) sin A + sin B = 2 sin( A+B2) cos( A B2) = 2 ...

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Standard XII

Mathematics

Circular Measurement of Angle

Prove that: s...

Question

Prove that: $\frac{sin 5 x + sin 3 x}{cos 5 x + cos 3 x} = tan 4 x$

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Solution

$\frac{sin 5 x + sin 3 x}{cos 5 x + cos 3 x} = tan 4 x$
$L . H . S = \frac{sin 5 x + sin 3 x}{cos 5 x + cos 3 x}$
$cos A + cos B = 2 cos (\frac{A + B}{2}) cos (\frac{A - B}{2})$
$sin A + sin B = 2 sin (\frac{A + B}{2}) cos (\frac{A - B}{2})$
$= \frac{2 sin (\frac{5 x + 3 x}{2}) cos (\frac{5 x - 3 x}{2})}{2 cos (\frac{5 x + 3 x}{2}) cos (\frac{5 x - 3 x}{2})} = \frac{2 sin (4 x) cos (x)}{2 cos (4 x) cos (x)}$
$= tan 4 x = R . H . S$

Hence proved.

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Similar questions

Q. Prove that:

\frac{sin x - sin 3 x + sin 5 x - sin 7 x}{cos x - cos 3 x - cos 5 x + cos 7 x} = cot 2 x

\frac{sin 5 x + sin 3 x}{cos 5 x + cos 3 x} =

Q. What is

\frac{sin 5 x - sin 3 x}{cos 5 x + cos 3 x}

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\frac{s i n x - s i n 3 x + s i n 5 x - s i n 7 x}{c o s x - c o s 3 x - c o s 5 x + c o s 7 x} = c o t 2 x

Q. Let

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Circular Measurement of Angle

Standard XII Mathematics

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Prove that: sin5x+sin3xcos5x+cos3x=tan4x

sin5x+sin3xcos5x+cos3x=tan4x L.H.S=sin5x+sin3xcos5x+cos3x cosA+cosB=2cos(A+B2)cos(A−B2) sinA+sinB=2sin(A+B2)cos(A−B2) =2sin(5x+3x2)cos(5x−3x2)2cos(5x+3x2)cos(5x−3x2)=2sin(4x)cos(x)2cos(4x)cos(x) =tan4x=R.H.S Hence proved.

Prove that: $\frac{sin 5 x + sin 3 x}{cos 5 x + cos 3 x} = tan 4 x$