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 ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Properties Derived from Trigonometric Identities

Prove the fol...

Question

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

Open in App

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.

Suggest Corrections

344

Similar questions

Q. Prove that

\frac{c o s 4 x + c o s 3 x + c o s 2 x}{s i n 4 x + s i n 3 x + s i n 2 x} = c o t 3 x

\frac{cos 4 x + cos 3 x + cos 2 x}{sin 4 x + sin 3 x + sin 2 x}

is equal to -

Q. Solve the following equations:
(i)

\cos x + \cos 2 x + \cos 3 x = 0

(ii)

\cos x + \cos 3 x - \cos 2 x = 0

(iii)

\sin x + \sin 5 x = \sin 3 x

(iv)

\cos x \cos 2 x \cos 3 x = \frac{1}{4}

(v)

\cos x + \sin x = \cos 2 x + \sin 2 x

(vi)

\sin x + \sin 2 x + \sin 3 = 0

(vii)

\sin x + \sin 2 x + \sin 3 x + \sin 4 x = 0

(viii)

\sin 3 x - \sin x = 4 \cos^{2} x - 2

(ix)

\sin 2 x - \sin 4 x + \sin 6 x = 0

Q. Identity transformations of Trigonometric Expressions.
prove the following identities.

\frac{s i n 2 x - s i n 3 x - s i n 4 x}{c o s 2 x - c o s 3 x + c o s 4 x} = t a n 3 x .

Q. Let

y = \frac{cos x + cos 2 x + cos 3 x + cos 4 x + cos 5 x + cos 6 x + cos 7 x}{sin x + sin 2 x + sin 3 x + sin 4 x + sin 5 x + sin 6 x + sin 7 x},

then which of the following hold good?

Join BYJU'S Learning Program

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

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