Answer
$$-1$$
Work Step by Step
Apply the sum formula:
$\cos(A+B)=\cos(A)\cos(B) -\sin(A)\sin(B)$
Therefore,
$$\cos (\dfrac{5 \pi}{12}) \ \cos(\dfrac{7 \pi}{12}) -\sin (\dfrac{5 \pi}{12}) \ \sin(\dfrac{7 \pi}{12})\\
=\cos \ (\dfrac{5\pi}{12} +\dfrac{ 7 \pi}{12})\\
=\cos \ (\dfrac{12\pi}{12})\\
=\cos \pi \\=-1$$
