Using sum identities and half angle identity

\dfrac{\sin8\degree +\cos15\degree\sin7\degree }{\cos8\degree-\sin15\degree\sin7\degree }

=\dfrac{\sin(15\degree-7\degree )+\cos15\degree\sin7\degree }{\cos(15\degree-7\degree )-\sin15\degree\sin7\degree }

= \dfrac{\sin15\degree\cos7\degree }{\cos15\degree\cos7\degree }

=\dfrac{\sin15\degree}{\cos15\degree}

=\tan15\degree =\tan(\dfrac{30\degree }{2} )

=\dfrac{\sin30\degree }{1+\cos30\degree }

=\dfrac{\dfrac{1}{2} }{1+\dfrac{\sqrt{3} }{2} }

=2-\sqrt{3}