`cot(u + v)` Find the exact value of the trigonometric expression given that sin(u) = 5/13 and cos(v) = -3/5 (both u and v are in quadrant II.)

Given `sin(u)=5/13 , cos(v)=-3/5`

using pythagorean identity,

`sin^2(u)+cos^2(u)=1`

`(5/13)^2+cos^2(u)=1`

`cos^2(u)=1-25/169=(169-25)/169=144/169`

`cos(u)=sqrt(144/169)`

`cos(u)=+-12/13`

since u is in quadrant II,

`:.cos(u)=-12/13`

`sin^2(v)+cos^2(v)=1`

`sin^2(v)+(-3/5)^2=1`

`sin^2(v)+9/25=1`

`sin^2(v)=1-9/25=(25-9)/25=16/25`

`sin(v)=sqrt(16/25)`

`sin(v)=+-4/5`

since v is in quadrant II,

`:.sin(v)=4/5`

Now let's evaluate cot(u+v),

`cot(u+v)=cos(u+v)/sin(u+v)`

`=(cos(u)cos(v)-sin(u)sin(v))/(sin(u)cos(v)+cos(u)sin(v))`

`=((-12/13)(-3/5)-(5/13)(4/5))/((5/13)(-3/5)+(-12/13)(4/5))`

`=(36/65-20/65)/(-15/65-48/65)`

`=-16/63`