"This can be done by twice using the double angle formula for cos."
"We have:"
\qquad \qquad cos2x \ = \ cos^2x - sin^2 x; \qquad \quad \ color{blue}{"double angle formula for cos"}
\qquad \qquad \qquad \qquad \quad \ \ = \ cos^2x - ( 1 - cos^2x )
\qquad \qquad \qquad \qquad \quad \ \ = \ cos^2x - 1 + cos^2x
\qquad \qquad \qquad \qquad \quad \ \ = \ 2 cos^2x - 1.
"Thus:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad cos2x \ = \ 2 cos^2x - 1. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad (I)
"So now:"
\qquad \qquad cos4x \ = \ cos( 2 cdot (2 x ) ); \qquad \qquad \qquad \qquad \qquad \qquad \qquad color{blue}{"now let:" \qquad A = 2x"}
\qquad \qquad \qquad= \ cos( 2 A );
\qquad \qquad \qquad= \ 2 cos^2A - 1; \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ color{blue}{"by (I)"}
\qquad \qquad \qquad= \ 2 ( cosA )^2 - 1;
\qquad \qquad \qquad= \ 2 ( cos 2 x )^2 - 1; \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad color{blue}{"since:" \qquad A = 2x"}
\qquad \qquad \qquad= \ 2 ( 2 cos^2x - 1 )^2 - 1; \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ \ \ color{blue}{"by (I)"}
\qquad \qquad \qquad= \ 2 ( [ 2 cos^2x ]^2 - 2 [ 2 cos^2x ] + 1 ) - 1;
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad color{blue}{ "as:" qquad ( a + b )^2 = a^2 + 2 a b + b^2 }
\qquad \qquad \qquad= \ 2 ( 4 cos^4x - 4 cos^2x + 1 ) - 1;
\qquad \qquad \qquad= \ 8 cos^4x - 8 cos^2x + 2 - 1;
\qquad \qquad \qquad= \ 8 cos^4x - 8 cos^2x + 1.
"This is what we wanted to show !!"
"Summarizing, we have shown:"
\qquad \qquad \qquad \qquad \qquad \qquad cos4x \ = \ 8 cos^4x - 8 cos^2x + 1.