# Write the correct alternative in the following:

Question:

Write the correct alternative in the following:

$\frac{d^{20}}{d x^{20}}(2 \cos x \cos 3 x)=$

A. $2^{20}\left(\cos 2 x-2^{20} \cos 4 x\right)$

B. $2^{20}\left(\cos 2 x+2^{20} \cos 4 x\right)$

c. $2^{20}\left(\sin 2 x-2^{20} \sin 4 x\right)$

D. $2^{20}\left(\sin 2 x-2^{20} \sin 4 x\right)$

Solution:

Given:

Let $y=2 \cos x \cos 3 x$

$2 \cos A \cos B=\cos \left(\frac{A+B}{2}\right)+\cos \left(\frac{A-B}{2}\right)$

So $y=\cos 2 x+\cos 4 x$

$\frac{d y}{d x}=-2 \sin 2 x-4 \sin 4 x$

$=(-2)^{1}\left(\sin 2 x+2^{1} \sin 4 x\right)$

$\frac{d^{2} y}{d x^{2}}=-4 \cos 2 x-16 \cos 4 x$

$=(-2)^{3}\left(\cos 2 x+2^{3} \cos 4 x\right)$

$\frac{\mathrm{d}^{3} \mathrm{y}}{\mathrm{dx}^{3}}=8 \sin 2 \mathrm{x}+64 \sin 4 \mathrm{x}$

$=(-2)^{3}\left(\cos 2 x+2^{3} \cos 4 x\right)$

$\frac{d^{4} y}{d x^{4}}=16 \cos 2 x+256 \cos 4 x$

$=(-2)^{4}\left(\cos 2 x+2^{4} \cos 4 x\right)$

For every odd degree; differential $==(-2)^{n}\left(\cos 2 x+2^{n} \cos 4 x\right) ; n=\{1,3,5 \ldots\}$

For every even degree; differential $=(-2)^{n}\left(\cos 2 x+2^{n} \cos 4 x\right) ; n=\{0,2,4 \ldots\}$

So, $\frac{\mathrm{d}^{20} \mathrm{y}}{\mathrm{dx}^{20}}=(-2)^{20}\left(\cos 2 \mathrm{x}+2^{20} \cos 4 \mathrm{x}\right)$

$=(-2)^{20}\left(\cos 2 x+2^{20} \cos 4 x\right)$