First pick a value of \(x\) close to \(\dfrac{12\pi}{13}\). You should be fine with \(x=\pi\).
The linear approximation of \(f(x)\) at \(x=c\) is given by
\(f(c)\approx f(a)+f'(a)(c-a)\)
where \(x=a\) is some fixed value close to \(x=c\). You have
\(f(x)=\cos x\implies f'(x)=-\sin x\)
so
\(f\left(\dfrac{12\pi}{13}\right)\approx f(\pi)+f'(\pi)\left(\dfrac{12\pi}{13}-\pi\right)\)
\(\cos\dfrac{12\pi}{13}\approx\cos\pi-\sin(\pi)\left(-\dfrac\pi{13}\right)\)
\(\cos\dfrac{12\pi}{13}\approx-1\)
The actual value is closer to -0.9709, so the approximation is decent.
