We will need the trig identity tan(A+B)=(tanA+tanB)/(1-tanAtanB).
We also need to remember that tan45=1 or 45=arctan(1).
Let the slope of L1 be m1 and the slope of L2 be m2, then:
arctan(m1)-arctan(m2)=45, the angle between the lines.
So arctan(m1)=45+arctan(m2) and:
m1=tan(45+arctan(m2))=tan(arctan(1)+arctan(m2)).
m2=-5. Remember that tan(arctan(x))=x.
Now apply the trig identity:
m1=(1-5)/(1+5)=-4/6=-⅔. So the slope of L1 is -⅔.