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    $\begingroup$ Welcome to Mathematica StackExchange! I suggest you read through some introductory tutorials, like Mathematica & Wolfram Language Fast Introduction for Math Students. You will learn there the basics, namely that you should use Sin instead of sin, and == for equations instead of = ... After that, please edit your question if you still encounter problems. $\endgroup$ Commented Jul 31 at 10:36
  • $\begingroup$ See mathematica.stackexchange.com/questions/18393/… for one "sin" you committed. :) $\endgroup$ Commented Jul 31 at 13:31
  • $\begingroup$ And mathematica.stackexchange.com/questions/11982/… for misused = (should be ==) $\endgroup$ Commented Jul 31 at 13:33
  • $\begingroup$ Ok let's correct the sins: Solve[{(Sin[a] + Sin[b])/Sin[a + b] == m, (Cos[a] + Cos[b])/Sin[a - b] == n}, {Sin[(a + b)/2]}] It still returns no result. The thing is I have solved this exercise manually on paper and I want to use Mathematica to verify it... the result I got is $$\frac{\pm \boldsymbol{mn}}{\sqrt{\boldsymbol{m}^{\boldsymbol{2}}\boldsymbol{n}^{\boldsymbol{2}}+\boldsymbol{1}}}$$ $\endgroup$ Commented Jul 31 at 14:08
  • $\begingroup$ @GJ Please check your equations. Latex and Mathematica code are different! $\endgroup$ Commented Jul 31 at 21:33