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    Spherical trigonometry proof

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    For a rectangular sperical triangle, where C = 90 degrees, show that:
    tan a = sin b tan A
    tan b = sin a tan B

    This is how far I have a gotten:
    sin C = 1
    cos C = 0

    I am also aware that I should be using both the Law of Cosines and the Law of Sines. Using these, I was able to prove several other identities. I am completely stuck on these two, however. Can someone help me?

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    Solution Preview

    See the attached file.

    Spherical Trigonometry deals with spherical triangles. On a sphere, great circles play the role of straight lines. The intersection of the sphere with a plane passing through its center gives a great circle. Line segments become great circle arcs. If the sphere has radius R, then all its great circles too will have the radius R; so the length of an arc is exactly R times the radian measure of the central angle it intersects or times the measure of that angle in degrees. Hence a great-circle arc, on the sphere, is the analogue of a straight line, on the plane.
    Where two such arcs intersect, we can define the spherical angle either as angle between the tangents to the two arcs, at the point of intersection, or as the ...

    Solution Summary

    The values for tan a and tan b are derived in the solution.