3.II.15G

Let $a, b, c$ be the lengths of a right-angled triangle in spherical geometry, where $c$ is the hypotenuse. Prove the Pythagorean theorem for spherical geometry in the form

$\cos c=\cos a \cos b$

Now consider such a spherical triangle with the sides $a, b$ replaced by $\lambda a, \lambda b$ for a positive number $\lambda$. Show that the above formula approaches the usual Pythagorean theorem as $\lambda$ approaches zero.

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