Precalculus, Chapter 5, 5.4, Section 5.4, Problem 43

Given sin(u)=5/13,cos(v)=-3/5
Angles u and v are in quadrant 2.
A right triangle can be drawn in quadrant 2. Since sin(u)=5/13 we know that the side opposite angle u is 5 and the hypotenuse of the triangle is 13. Using the pythagorean theorem the third side of the triangle is 12.
A second triangle can be drawn in quadrant 2. Since cos(v)=-3/5 we know that the side adjacent angle v is 3 and the hypotenuse of the triangle is 5. Using the pythagorean theorem the third side of the triangle is 4.
tan(u+v)=(tan(u)+tan(v))/(1-tan(u)tan(v))
tan(u+v)=((-5/12)+(-4/3))/(1-(-5/12)(-4/3))=((-5+(-16))/12)/(1-(20/36))=(-21/12)/(16/36)=-63/16