Prove the following-cos 4 x-cos 3 x-cos 2 x-sin 4 x-sin 3 x-sin 2 x- 3 x

We have L.H.S. = cos 4x+ cos 3x+ cos 2x/sin 4x + sin 3x+ sin 2x = (cos 4x+ cos 2x)+ cos 3x/(sin 4x + sin 2x)+ sin 3x = 2 cos4x+2x/2cos4x 2x/2+ cos 3x/2 sin4x+2 ...

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Byju's Answer

Standard XII

Mathematics

Properties Derived from Trigonometric Identities

Prove the fol...

Question

Prove the following:
$\frac{cos 4x+ cos 3x + cos 2x}{sin 4x + sin 3x + sin 2x} = cot 3 x$

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Solution

We have

L.H.S. = $\frac{cos 4x+ cos 3x+ cos 2x}{sin 4x + sin 3x+ sin 2x}$

= $\frac{(cos 4x+ cos 2x)+ cos 3x}{(sin 4x + sin 2x)+ sin 3x}$

= $\frac{2 cos \frac{4 x + 2 x}{2} cos \frac{4 x - 2 x}{2} + cos 3x}{2 sin \frac{4 x + 2 x}{2} cos \frac{4 x - 2 x}{2} + sin 3x}$

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

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

= $\frac{cos 3x (2 cos x+1)}{sin 3x (2 cos x +1)} = \frac{cos 3x}{sin 3x}$

= cot 3x = R.H.S.

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is equal to -

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Prove the following: cos 4x+ cos 3x + cos 2xsin 4x + sin 3x + sin 2x=cot 3x

Prove the following:
$\frac{cos 4x+ cos 3x + cos 2x}{sin 4x + sin 3x + sin 2x} = cot 3 x$