If the period of the function f(x) = sin 5x c | vedclass

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(D) Given the function $f(x) = \sin 5x \cos 3x$. Using the trigonometric identity $2 \sin A \cos B = \sin(A+B) + \sin(A-B)$, we can rewrite the functi

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(D) Given the function $f(x) = \sin 5x \cos 3x$. Using the trigonometric identity $2 \sin A \cos B = \sin(A+B) + \sin(A-B)$, we can rewrite the function as: $f(x) = \frac{1}{2} [\sin(5x+3x) + \sin(5x-3x)] = \frac{1}{2} (\sin 8x + \sin 2x)$. The period of $\sin 8x$ is $T_1 = \frac{2\pi}{8} = \frac{\pi}{4}$. The period of $\sin 2x$ is $T_2 = \frac{2\pi}{2} = \pi$. The period of the sum of two periodic functions is the Least Common Multiple ($L$.$C$.$M$.) of their individual periods. $\alpha = 	ext{L.C.M.}\left(\frac{\pi}{4}, \piight) = \frac{	ext{L.C.M.}(\pi, \pi)}{	ext{H.C.F.}(4, 1)} = \frac{\pi}{1} = \pi$. Thus, $\alpha = \pi$. Finally, $\cos \alpha = \cos \pi = -1$.

If the period of the function $f(x) = \sin 5x \cos 3x$ is $\alpha$, then $\cos \alpha =$

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