Question

Find the period of the following function

${4cos}^{4} ( \frac{x - \pi}{ {4\pi}^{2} }) - 2cos( \frac{x - \pi}{ {2\pi}^{2} })$
​

Answer 1

$\large\underline{\sf{Solution-}}$

Given function is

$\rm \: f(x) = 4{cos}^{4}\bigg(\dfrac{x - \pi}{ {4\pi}^{2} } \bigg) - 2cos\bigg(\dfrac{x - \pi}{ {2\pi}^{2} } \bigg)$

can be further rewritten as

$\rm \: f(x) = \bigg[2{cos}^{2}\bigg(\dfrac{x - \pi}{ {4\pi}^{2} } \bigg)\bigg]^{2} - 2cos\bigg(\dfrac{x - \pi}{ {2\pi}^{2} } \bigg)$

We know,

$\boxed{\tt{ \: \: {2cos}^{2}x = 1 + cos2x \: \: }}$

So, using this, we get

$\rm \: f(x) = {\bigg[1 + cos2\bigg(\dfrac{x - \pi}{ {4\pi}^{2} } \bigg) \bigg]}^{2} - 2cos\bigg(\dfrac{x - \pi}{ {2\pi}^{2} } \bigg)$

$\rm \: f(x) = {\bigg[1 + cos\bigg(\dfrac{x - \pi}{ {2\pi}^{2} } \bigg) \bigg]}^{2} - 2cos\bigg(\dfrac{x - \pi}{ {2\pi}^{2} } \bigg)$

$\rm \: f(x) = 1 + {cos}^{2}\bigg(\dfrac{x - \pi}{ {2\pi}^{2} } \bigg) + cos\bigg(\dfrac{x - \pi}{ {2\pi}^{2} } \bigg) - 2cos\bigg(\dfrac{x - \pi}{ {2\pi}^{2} } \bigg)$

$\rm \: f(x) = 1 + {cos}^{2}\bigg(\dfrac{x - \pi}{ {2\pi}^{2} } \bigg)$

$\rm \: f(x) = \dfrac{1}{2}\bigg[2 + 2{cos}^{2}\bigg(\dfrac{x - \pi}{ {2\pi}^{2} } \bigg)\bigg]$

Again, using the result,

$\boxed{\tt{ \: \: {2cos}^{2}x = 1 + cos2x \: \: }}$

we get

$\rm \: f(x) = \dfrac{1}{2}\bigg(2 + 1 + cos2\bigg(\dfrac{x - \pi}{ {2\pi}^{2} } \bigg) \bigg)$

$\rm \: f(x) = \dfrac{1}{2}\bigg(3 + cos\bigg(\dfrac{x - \pi}{ {\pi}^{2} } \bigg) \bigg)$

can be further rewritten as

$\rm \: f(x) = \dfrac{1}{2}\bigg(3 + cos\bigg(\dfrac{x}{ {\pi}^{2}} - \dfrac{\pi}{ {\pi}^{2} } \bigg) \bigg)$

$\rm \: f(x) = \dfrac{1}{2}\bigg(3 + cos\bigg(\dfrac{x}{ {\pi}^{2}} - \dfrac{1}{\pi} \bigg) \bigg)$

We know,

$\boxed{\sf{ Period\:of\:cos(ax + b) = \: \frac{2\pi}{a} }}$

So, using this result, we have

$\rm \: Period\:of\:f(x) =\dfrac{2\pi}{\dfrac{1}{ {\pi}^{2} } }$

$\rm\implies \: \: \boxed{\tt{ \: \rm \: Period\:of\:f(x) = {2\pi}^{3} \: \: }}$

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ADDITIONAL INFORMATION

$\begin{gathered}\boxed{\begin{array}{c|c} \bf Function & \bf Period \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf sinx, \: cosx & \sf 2\pi \\ \\ \sf tanx, \: cotx & \sf \pi \\ \\ \sf |sinx|, \: |cosx|, \: |tanx| & \sf \pi\\ \\ \sf {sin}^{n}x, \: {cos}^{n}x & \sf \pi \: if \: n \: is \: even\\ \\ \sf {sin}^{n}x, \: {cos}^{n}x & \sf 2\pi \: if \: n \: is \: odd\\ \\ \sf sin(ax + b), \: cos(ax + b) & \sf \dfrac{2\pi}{a} \end{array}} \\ \end{gathered}$