Write the value

We know that

$\cos ^{-1}(\cos x)=x$

Now,

$\cos ^{-1}\left(\cos 1540^{\circ}\right)=\cos ^{-1}\left\{\cos \left(1440+100^{\circ}\right)\right\}$

$=\cos ^{-1}\left\{\cos \left(100^{\circ}\right)\right\} \quad\left[\because \cos \left(4 \pi+100^{\circ}\right)=\cos 100^{\circ}\right]$

$=100^{\circ}$