Question 23
Let the numbers $\alpha, \beta$ satisfy $0\lt \alpha\lt \beta\lt \pi / 2$ and let $\gamma$ and $\delta$ be the numbers defined by the conditions:
(i) $0\lt \gamma\lt \pi / 2$, and $\tan \gamma$ is the arithmetic mean of $\tan \alpha$ and $\tan \beta$;
(ii) $0\lt \delta\lt \pi / 2$, and $\frac{1}{\cos \delta}$ is the arithmetic mean of $\frac{1}{\cos \alpha}$ and $\frac{1}{\cos \beta}$.
Prove that $\gamma\lt \delta$.
(i) $0\lt \gamma\lt \pi / 2$, and $\tan \gamma$ is the arithmetic mean of $\tan \alpha$ and $\tan \beta$;
(ii) $0\lt \delta\lt \pi / 2$, and $\frac{1}{\cos \delta}$ is the arithmetic mean of $\frac{1}{\cos \alpha}$ and $\frac{1}{\cos \beta}$.
Prove that $\gamma\lt \delta$.