How to simplify $\sin^4 x+\cos^4 x$ using trigonometrical identities? $\sin^{4}x+\cos^{4}x$
I should rewrite this expression into a new form to plot the function.
\begin{align}
& = (\sin^2x)(\sin^2x) - (\cos^2x)(\cos^2x) \\
& = (\sin^2x)^2 - (\cos^2x)^2 \\
& = (\sin^2x - \cos^2x)(\sin^2x + \cos^2x) \\
& = (\sin^2x - \cos^2x)(1) \longrightarrow\,
= \sin^2x - \cos^2x
\end{align}
Is that true?

Note that $a^2 + b^2 = (a+b)^2 - 2ab$
$$(\sin^2 x)^2 + (\cos^2 x)^2 = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x\cos^2 x =(\sin^2 x + \cos^2 x)^2 - 2(\sin x\cos x)^2 = \\ 1 -\frac{ \sin^2 2x}{2}$$
Note the following results: 
$$ \sin^2 x + \cos^2 x = 1$$
$$ \sin x \cos x = \frac{\sin 2x}{2}$$
