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  4. The fundamental period of the function f(x)=4 cos 4((x-π/4 π2))-2 cos ((x-π/2 π2)) is equal to

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Q. The fundamental period of the function $f(x)=4 \cos ^4\left(\frac{x-\pi}{4 \pi^2}\right)-2 \cos \left(\frac{x-\pi}{2 \pi^2}\right)$ is equal to

Relations and Functions - Part 2

Solution:

We have, $f(x)=4 \cos ^4\left(\frac{x-\pi}{4 \pi^2}\right)-2 \cos \left(\frac{x-\pi}{2 \pi^2}\right)$
[13th, 31-07-2011, P-1]
$=\left(2 \cos ^2\left(\frac{x-\pi}{4 \pi^2}\right)\right)^2-2 \cos \left(\frac{x-\pi}{2 \pi^2}\right)=\left(1+\cos \left(\frac{x-\pi}{2 \pi^2}\right)\right)^2-2 \cos \left(\frac{x-\pi}{2 \pi^2}\right)=1+\cos ^2\left(\frac{x-\pi}{2 \pi^2}\right)$
Clearly, period of $f ( x )=\frac{\pi}{\frac{1}{2 \pi^2}}=2 \pi^3$.