Find the value of frac{d^{20}}{dx^{20}}(2cos | vedclass

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(B) Given expression is $y = 2\cos x \cos 3x$. Using the trigonometric identity $2\cos A \cos B = \cos(A+B) + \cos(A-B)$,we have: $y = \cos(3x+x) + \c

Vedclass · Accepted Answer

$2^{20}(\cos 2x + 2^{20}\cos 4x)$

Vedclass · Answer

(B) Given expression is $y = 2\cos x \cos 3x$. Using the trigonometric identity $2\cos A \cos B = \cos(A+B) + \cos(A-B)$,we have: $y = \cos(3x+x) + \cos(3x-x) = \cos 4x + \cos 2x$. Now,we need to find the $20^{th}$ derivative of $y$ with respect to $x$. Recall that $\frac{d^n}{dx^n}(\cos(ax)) = a^n \cos(ax + \frac{n\pi}{2})$. For $n=20$,$\frac{d^{20}}{dx^{20}}(\cos(ax)) = a^{20} \cos(ax + \frac{20\pi}{2}) = a^{20} \cos(ax + 10\pi) = a^{20} \cos(ax)$. Applying this to our expression: $\frac{d^{20}}{dx^{20}}(\cos 4x + \cos 2x) = 4^{20} \cos 4x + 2^{20} \cos 2x$. Since $4^{20} = (2^2)^{20} = 2^{40}$,the expression becomes $2^{40} \cos 4x + 2^{20} \cos 2x$. Factoring out $2^{20}$,we get $2^{20}(2^{20} \cos 4x + \cos 2x)$. This matches option $B$.

Find the value of $\frac{d^{20}}{dx^{20}}(2\cos x \cos 3x)$.

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