Answer
$\theta_{max}=42^o$
Work Step by Step
According to Newton's 3rd Law of Motion, if $F$ is the force exerted on each crutch by the ground, there has to be a $P$ force in equal magnitude but in opposite direction exerted on the ground by the crutch. We will be concerned with force $P$ here, as portrayed in the free-body diagram below.
$\vec{P}$, in the diagram, is analyzed $\vec{P}\cos\theta$ and $\vec{P}\sin\theta$.
- $\vec{P}\sin\theta$ is the force that causes the crutch to slip, if it overcomes frictional force $f_s^{max}$. Since we are finding $\theta_{max}$ so that the crutch does not slip, we take that $$P\sin\theta_{max}=f_s^{max}=\mu_sF_N$$
- On the vertical sides, we have $\vec{P}\cos\theta$ and $\vec{F}_N$ in opposite directions. Since there is no vertical movement, we have $$F_N=P\cos\theta$$
Therefore, $$P\sin\theta_{max}=\mu_sP\cos\theta_{max}$$ $$\frac{\sin\theta_{max}}{\cos\theta_{max}}=\tan\theta_{max}=\mu_s=0.9$$ $$\theta_{max}=42^o$$
