$\cos \left(2 x\right)$

$\= {\cos }^2\left(x\right)-{\sin }^2\left(x\right)$

$\= 2 {\cos }^2\left(x\right)-1$

$\= 1-2 {\sin }^2\left(x\right)$

$\cos \left(2\cdot \left(2 x\right)\right)$

$\= 2 {\cos }^2\left(2 x\right)-1$

$\= 2{\left(2 {\cos }^2\left(x\right)-1\right)}^2-1$

$\= 2\left(4 {\cos }^4\left(x\right)-4 {\cos }^2\left(x\right)\+1\right)-1$

$\= 8 {\cos }^4\left(x\right)-8 {\cos }^2\left(x\right)\+1$

$\cos \left(4x\right)+4\cos \left(2x\right)+3$

$\=8 {\cos }^4\left(x\right)-8 {\cos }^2\left(x\right)\+1\+4\left(2 {\cos }^2\left(x\right)-1\right)\+3$

$\=8 {\cos }^4\left(x\right)-8 {\cos }^2\left(x\right)\+8 {\cos }^2\left(x\right)\+1-4\+3$

$\=8 {\cos }^4\left(x\right)$

$\cos \left(4x\right)+4\cos \left(2x\right)+3$$\stackrel{\text{expand}}{\=}$$8{\cos \left(x\right)}^4$

Stepwise expansions

$\cos \left(4 x\right)$$\stackrel{\text{expand}}{\=}$$8{\cos \left(x\right)}^4-8{\cos \left(x\right)}^2\+1$

$4\cos \left(2x\right)$$\stackrel{\text{expand}}{\=}$$8{\cos \left(x\right)}^2-4$

$\mathrm{q\_\_1}≔\cos \left(4x\right)+4\cos \left(2x\right)+3\:$

$\mathrm{expand}\left(\mathrm{q\_\_1}\right)$

$8{\cos \left(x\right)}^4$

Stepwise expansions

$\mathrm{q\_\_2}≔\mathrm{expand}\left(\mathrm{op}\left(1\,\mathrm{q\_\_1}\right)\right)$

$8{\cos \left(x\right)}^4-8{\cos \left(x\right)}^2\+1$

$\mathrm{q\_\_3}≔ \mathrm{expand}\left(\mathrm{op}\left(2\,\mathrm{q\_\_1}\right)\right)$

$8{\cos \left(x\right)}^2-4$

Sum the sub-expressions

$\mathrm{q\_\_2}\+\mathrm{q\_\_3}\+3$

$8{\cos \left(x\right)}^4$