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\LARGE
\textcolor{blue}
{\begin{eqnarray*}
\sin\theta_c &=&\frac{\tan\theta_c}{\sqrt{1+\tan^2\theta_c}}
=\frac{\mu_k}{\sqrt{1+\mu_k^2}}\\
\cos\theta_c &=&\frac{1}{\sqrt{1+\tan^2\theta_c}}=\frac{1}{\sqrt{1+\mu_k^2}}\\
\end{eqnarray*}
}

\noindent The final result is:
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{\begin{eqnarray*}
ma_{\rm max} &=& F(\mu_k\sin\theta_c+\cos\theta_c)-\mu_k mg\\
 &=&F\left(\frac{\mu_k^2}{\sqrt{1+\mu_k^2}}+\frac{1}{\sqrt{1+\mu_k^2}}\right)-\mu_k mg\\
&=&F\left(\frac{1+ \mu_k^2}{\sqrt{1+\mu_k^2}}\right)-\mu_k mg
\end{eqnarray*}
}
\end{document}
