Is $\cos(x^2)$ the same as $\cos^2(x)$? I want to know something about trigonometrical functions, is $\cos(x^2)$ the same as $\cos^2(x)$ ?
One of the first things you may observe is that $(\cos {\small{(}}x{\small{)}})^2\geqslant0$ whereas $\cos(x^2)$ may be equal to a negative number. (Why?)

In blue, a graph of the function $\color{blue}{\cos^2(x)}$ and in red a graph of the function $\color{red}{\cos(x^2)}$.



Some say, a good plot is worth a million words! :-)
No. $\cos x^2=\cos(x^2)=\cos(x\times x)$ while $\cos^2(x)=\cos(x)\times \cos(x)=(\cos(x))^2$.