BALTIC-WAY 1998 Problem 9

Let the numbers $\alpha, \beta$ satisfy $0<\alpha<\beta<\pi / 2$ and let $\gamma$ and $\delta$ be the numbers defined by the conditions:

(i) $0<\gamma<\pi / 2$, and $\tan \gamma$ is the arithmetic mean of $\tan \alpha$ and $\tan \beta$;

(ii) $0<\delta<\pi / 2$, and $\frac{1}{\cos \delta}$ is the arithmetic mean of $\frac{1}{\cos \alpha}$ and $\frac{1}{\cos \beta}$.

Prove that $\gamma<\delta$.