What Formula Connects Cosine To Tangent?

Relevant wiki: Fundamental Trigonometric Identities - Problem Solving - Medium

Let's eliminate the variables one-by-one. From the first two equations and using the identity that $\sec^2 \beta = 1+\tan^2\beta$ we have, $\cot^2\gamma=1+\cos^2\alpha$ Using the third equation, we have $\cot^2\gamma=1+\frac{1}{1+\cos^2\gamma}$ Letting $\sin \gamma =x$ , we obtain the following algebraic equation from the above $\frac{1}{x^2}=2+\frac{1}{2-x^2}$ which is a quadratic equation in $x^2$ and can be readily solved. Using the constraint that $\gamma$ is an acute angle, we find the only acceptable solution is $x=\sin \gamma = \frac{\sqrt{5}-1}{2}$

The symmetry of the problem suggests that there may exist a solution with $\alpha = \beta = \gamma$ . Let's give it a try:

$\cos \gamma = \tan \gamma \\ \cos^2 \gamma = \sin \gamma \\ \sin^2 \gamma + \sin \gamma - 1 = 0 \\ (\sin \gamma + \tfrac 12)^2 = \tfrac 54 \\ \sin \gamma = -\tfrac12 + \tfrac12\sqrt 5 \approx \boxed{0.62}.$