For starters,
[tex]\cos\dfrac{7\pi}4=\cos\left(2\pi-\dfrac\pi4\right)=\cos\left(-\dfrac\pi4\right)=\cos\dfrac\pi4=\dfrac1{\sqrt2}[/tex]
The second equality follows from the fact that [tex]\cos x[/tex] has a period of [tex]2\pi[/tex], and the equality after that follows from the parity of [tex]\cos x[/tex] (i.e. [tex]\cos(-x)=\cos x[/tex]).
Then
[tex]\cos^{-1}\left(\cos\dfrac{7\pi}4\right)=\cos^{-1}\left(\cos\dfrac\pi4\right)=\cos^{-1}\dfrac1{\sqrt2}=\boxed{\dfrac\pi4}[/tex]
