Answer
See the detailed answer below.
Work Step by Step
When the diver needs to increase his angular speed, he needs to decrease his body moment of inertia so he can make 3 rotations at the same time interval rather than one and a half rotations.
We can use the conservation of angular momentum here.
$$I_1\omega_1=I_2\omega_2$$
where $I_1$ refers to the pike position and $I_2$ refers to the tuck position.
$$\dfrac{\omega_2}{\omega_1}=\dfrac{I_1}{I_2}$$
When we need $\omega_2\gt \omega_1$, then $I_1/I_2$ must be greater than 1.
This means that $I_1\gt I_2$.
So he has to decrease his moment of inertia by using the tuck position so he can rotate faster.