sin θ_{c} = \frac{tan θ_{c}}{\sqrt{1 + tan^{2} θ_{c}}} = \frac{μ}{\sqrt{1 + μ^{2}}}

cos θ_{c} = \frac{1}{\sqrt{1 + tan^{2} θ_{c}}} = \frac{1}{\sqrt{1 + μ^{2}}}

The final result is:

a_{max} = F (μ sin θ_{c} + cos θ_{c}) − μmg
= F (\frac{μ^{2}}{\sqrt{1 + μ^{2}}} + \frac{1}{\sqrt{1 + μ^{2}}} ) − μmg
= F (\frac{1 + μ^{2}}{\sqrt{1 + μ^{2}}} ) − μmg