Calculus Level 5

A radioactive source (which can be assumed to be a point source) is situated a distance of 1 metre from a large photographic plate (which can be assumed to be an infinite plane). There is a target area on the plate, a square of side 1 metre whose centre is the point on the plate which is closest to the source.

When a radioactive particle is emitted from the source, the direction in which it is emitted is totally random. This means that the probability that an emitted particle hits a particular region on the photographic plate is proportional to the solid angle subtended by that region of the plate at the source. A radioactive particle that is emitted from the source travels in a straight line: the effects of gravity and other forces may be ignored.

The probability that an emitted particle which hits the photographic plate hits the target area can be shown to be equal to $1 - \frac{A}{\pi^B} \tan^{-1}\left(\sqrt{\tfrac{C}{D}}\right)$ where $$A,B,C,D$$ are positive integers, with $$C,D$$ distinct primes. What is the value of $$A+B+C+D$$?

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