$\begin{array}{l}
\sin \left( {x + {{24}^o}} \right) + \cos \left( {x + {{144}^o}} \right) = \sin {15^o}\\
\Leftrightarrow \sin \left( {x + {{24}^o}} \right) + \cos \left( {{{90}^o} + {{54}^o} + x} \right) = \sin {15^o}\\
\Leftrightarrow \sin \left( {x + {{24}^o}} \right) - \sin \left( {{{54}^o} + x} \right) = \sin {15^o}\\
\Leftrightarrow 2\cos \left( {x + {{39}^o}} \right).\sin \left( { - {{15}^o}} \right) = \sin {15^o}\\
\Leftrightarrow - 2\cos \left( {x + {{39}^o}} \right).\sin {15^o} = \sin {15^o}\\
\Leftrightarrow \cos \left( {x + {{39}^o}} \right) = - \dfrac{1}{2}\\
\Leftrightarrow \left[ \begin{array}{l}
x + {39^o} = {120^o} + k{360^o}\\
x + {39^o} = - {120^o} + k{360^o}
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = {81^o} + k{360^o}\\
x = - {159^o} + k{360^o}
\end{array} \right.\left( {k \in Z} \right)
\end{array}$
