\[\begin{array}{l}
Cos\left( {4x + \frac{\pi }{3}} \right) = sin\left( {x + \frac{\pi }{5}} \right)\\
\Leftrightarrow Cos\left( {4x + \frac{\pi }{3}} \right) = \cos \left( {\frac{\pi }{2} - x - \frac{\pi }{5}} \right)\\
\Leftrightarrow Cos\left( {4x + \frac{\pi }{3}} \right) = \cos \left( {\frac{{3\pi }}{{10}} - x} \right)\\
\Leftrightarrow \left[ \begin{array}{l}
4x + \frac{\pi }{3} = \frac{{3\pi }}{{10}} - x + k2\pi \\
4x + \frac{\pi }{3} = \frac{{ - 3\pi }}{{10}} + x + k2\pi
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = \frac{{ - \pi }}{{150}} + \frac{{k2\pi }}{5}\\
x = \frac{{ - 19\pi }}{{90}} + \frac{{k2\pi }}{3}
\end{array} \right.\left( {k \in Z} \right)\\
vay:S = \{ \frac{{ - \pi }}{{150}} + \frac{{k2\pi }}{5};\frac{{ - 19\pi }}{{90}} + \frac{{k2\pi }}{3}\}
\end{array}\]

Cos(4x+π3)=sin(x+π5)⇔Cos(4x+π3)=cos(π2−x−π5)⇔Cos(4x+π3)=cos(3π10−x)⇔[4x+π3=3π10−x+k2π4x+π3=−3π10+x+k2π⇔[x=−π150+k2π5x=−19π90+k2π3(k∈Z)vay:S={−π150+k2π5;−19π90+k2π3}