Question 1074974
{{{sin^2(u)=64/289}}}
{{{cos^2(u)=1-sin^2(u)}}}
{{{cos^2(u)=1-64/289}}}
{{{cos^2(u)=289/289-64/289}}}
{{{cos^2(u)=225/289}}}
{{{cos(u)=15/17}}} or {{{cos(u)=-15/17}}}
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Similarly,
{{{cos^2(v)=81/2209}}}
{{{sin^2(v)=1-cos^2(v)}}}
{{{sin^2(v)=1-81/2209}}}
{{{sin^2(v)=2209/2209-81/2209}}}
{{{sin^2(v)=2128/2209}}}
{{{sin(v)=sqrt(2128)/47}}} or {{{sin(v)=-sqrt(2128)/47}}}
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So then,
{{{csc(u-v)=1/sin(u-v)}}}
and
{{{sin(u-v)=sin(u)cos(v)-sin(v)cos(u)}}}
So there are 4 possible solutions.
1. {{{cos(u)}}} positive, {{{sin(v)}}} positive
2. {{{cos(u)}}} positive, {{{sin(v)}}} negative
3. {{{cos(u)}}} negative, {{{sin(v)}}} positive
4. {{{cos(u)}}} negative, {{{sin(v)}}} negative

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I'll do case #1, you do the others the same way.
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{{{sin(u-v)=sin(u)cos(v)-sin(v)cos(u)}}}
{{{sin(u-v)=(8/17)(-9/47)-(sqrt(2128)/47)(15/17)}}}
{{{sin(u-v)=(-72/799)-(15/799)(sqrt(2128))}}}
{{{sin(u-v)=-(72+15sqrt(2128))/799}}}
So then,
{{{csc(u-v)=-799/(72+15sqrt(2128))}}}