`csc(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)=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 csc(u-v),

`csc(u-v)=1/sin(u-v)`

`=1/(sin(u)cos(v)-cos(u)sin(v))`

plug in the values of sin(u),cos(v),cos(u) and sin(v),

`=1/((5/13)(-3/5)-(-12/13)(4/5))`

`=1/(-3/13+48/65)`

`=65/33`