Prove the following-sin 5 x-sin 3 x-cos 5 x-cos 3 x-tan 4 x

We have L.H.S. = sin 5x + sin 3x/cos 5x + cos 3x = 2 sin (5x+3x/2) cos(5x 3x/2)/2 cos (5x+3x/2) cos(5x 3x/2) [ ∵sin C+sin D= 2 sin ( C+D/2) cos (C D/2); cos ...

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Byju's Answer

Standard XI

Mathematics

Trigonometric Ratios of Multiples of an Angle

Prove the fol...

Question

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

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Solution

We have

L.H.S. = $\frac{sin 5x + sin 3x}{cos 5x + cos 3x}$

= $\frac{2 sin (\frac{5 x + 3 x}{2}) cos (\frac{5 x - 3 x}{2})}{2 cos (\frac{5 x + 3 x}{2}) cos (\frac{5 x - 3 x}{2})}$

$⎡ ⎢ ⎣ \begin{matrix} ∵ sin C + sin D = 2 sin (\frac{C + D}{2}) cos (\frac{C - D}{2}) cos C + cos D = 2 cos (\frac{C + D}{2}) cos (\frac{C - D}{2}) \end{matrix} ⎤ ⎥ ⎦$

= $\frac{2 sin 4x cos x}{2 cos 4x cos x} = tan$ 4x

= R.H.S.

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Similar questions

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\frac{(sin 7 x + sin 5 x) + (sin 9 x + sin 3 x)}{(cos 7 x + cos 5 x) + (cos 9 x + cos 3 x)} = tan 6 x

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\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}

equal to?

\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

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Prove the following: sin 5x + sin 3xcos 5x + cos 3x=tan 4x

We have L.H.S. = sin 5x + sin 3xcos 5x + cos 3x = 2 sin(5x+3x2) cos(5x−3x2)2 cos(5x+3x2)cos(5x−3x2) ⎡⎢⎣∵sin C+sin D=2sin(C+D2)cos(C−D2)cos C + cos D=2cos(C+D2)cos(C−D2)⎤⎥⎦ = 2 sin 4x cos x2 cos 4x cos x=tan 4x = R.H.S.

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