How can the angle between any two azimuth dip pairs be found?

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The order of rotation is azimuth then dip. I've tried pythagoras style answers but testing showed it was the wrong approach. How can I find the angle between any two pairs?

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I suggest you turn your spherical coordinates into cartesian ones of unit length. Then the dot product between the vectors can be computed easily, and it is equal to the cosine of the angle.