Answer
$$\frac{1}{2}$$
Work Step by Step
$$\eqalign{
& \int_0^{\pi /4} {\int_0^{\pi /4} {\int_0^{\cos \phi } {\cos \theta } d\rho d\phi d\theta } } \cr
& \int_0^{\pi /4} {\int_0^{\pi /4} {\left[ {\int_0^{\cos \phi } {\cos \theta } d\rho } \right]d\phi d\theta } } \cr
& {\text{Integrate with respect to }}\rho \cr
& \int_0^{\cos \phi } {\cos \theta } d\rho = \cos \theta \left[ \rho \right]_0^{\cos \phi } \cr
& = \cos \theta \cos \phi \cr
& \int_0^{\pi /4} {\int_0^{\pi /4} {\left[ {\int_0^{\cos \phi } {\cos \theta } d\rho } \right]d\phi d\theta } } = \int_0^{\pi /4} {\int_0^{\pi /4} {\cos \theta \cos \phi d\phi d\theta } } \cr
& {\text{Integrate with respect to }}\phi \cr
& = \int_0^{\pi /4} {\cos \theta \left[ {\sin \phi } \right]_0^{\pi /4}d\theta } \cr
& = \int_0^{\pi /4} {\cos \theta \left( {\frac{{\sqrt 2 }}{2}} \right)d\theta } \cr
& = \frac{{\sqrt 2 }}{2}\int_0^{\pi /4} {\cos \theta d\theta } \cr
& {\text{Integrate}} \cr
& = \frac{{\sqrt 2 }}{2}\left[ {\sin \theta } \right]_0^{\pi /4} \cr
& = \frac{{\sqrt 2 }}{2}\left( {\frac{{\sqrt 2 }}{2}} \right) \cr
& = \frac{1}{2} \cr} $$
