What is the value of $an left(frac{5 pi}{6} ight) sin left(frac{7 pi}{6} ight) ?$

$an left(frac{5 pi}{6} ight) cdot sin left(frac{7 pi}{6} ight)= an left(pi-frac{pi}{6} ight) cdot sin left(pi+frac{pi}{6} ight)$

[ tan (180° – θ) = –tanθ

sin (180° + θ) = –sinθ]

$=- an frac{pi}{6} cdotleft(-sin frac{pi}{6} ight)$

$= an frac{pi}{6} cdot sin frac{pi}{6}=frac{1}{sqrt{3}} cdot frac{1}{2}=frac{1}{2 sqrt{3}}$