Question

Express $\cos \left(4\theta \right)$ in terms of $\cos \left(\theta \right)$.

Answer 1

Apply the identity of $\cos \left(2x\right)$ in terms of $\cos \left(x\right)$

Now, as we know the identity $\cos \left(2x\right)=2{\cos }^2\left(x\right)-1$

Substitute $x=2\theta$ we have $\cos \left(4\theta \right)=2{\cos }^2\left(2\theta \right)-1$

Again using the above-used identity,

$$\begin{gathered} \cos \left(4\theta \right)=2{\left[2{\cos }^2\left(\theta \right)-1)\right]}^2-1 \\ \cos \left(4\theta \right)=2\left[{{\left\{2{\cos }^2\left(\theta \right)\right\}}^2+1}^2-2\left(2{\cos }^2\right(\theta \left)\right)1\right]-1 \\ \cos \left(4\theta \right)=2\left[4{\cos }^4\left(\theta \right)+1-4{\cos }^2\theta \right]-1 \\ \cos \left(4\theta \right)=8{\cos }^4\left(\theta \right)+2-8{\cos }^2\theta -1 \\ \cos \left(4\theta \right)=8{\cos }^4\left(\theta \right)-8{\cos }^2\theta +1 \end{gathered}$$

Hence, the answer is $\cos \left(4\theta \right)=8{\cos }^4\left(\theta \right)-8{\cos }^2\theta +1$