Ans: D
I may be false. For $\theta= 40^\circ$, $\tan 40^\circ=0.8391$ and $\cos 40^\circ=0.7660$. $\tan 40^\circ > \cos 40^\circ$.
I may be false. For $\theta= 40^\circ$, $\tan 40^\circ=0.8391$ and $\cos 40^\circ=0.7660$. $\tan 40^\circ > \cos 40^\circ$.
II must be true. For $0^\circ < \theta < 45^\circ$,
$\begin{array}{cl}
& \sin \theta – \tan \theta \\
= & \sin \theta – \dfrac{\sin \theta}{ \cos \theta} \\
= & \sin \theta \times \left ( 1 – \dfrac{1}{\cos\theta} \right) \\
\end{array}$
Since $\sin\theta >0$ and $\dfrac{1}{\cos\theta}>1$ for $0^\circ < \theta <45^\circ$, then we have
$\begin{array}{rcl}
\sin \theta \times \left ( 1 – \dfrac{1}{\cos\theta} \right) & < & 0 \\
\sin \theta - \tan \theta & < & 0 \\
\sin \theta & < & \tan \theta
\end{array}$
III must be true. For $0^\circ <\theta <45^\circ$,
$\begin{array}{rcl}
\tan \theta & < & 1 \\
\dfrac{\sin\theta}{\cos\theta} & < & 1 \\
\sin \theta & < & \cos \theta
\end{array}$
