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Have unit square sides of 1. make Rotation it with angle λ.
Unit circle radius 1 with origin center is has point Q on the squares corner. is [ cos λ , sin λ ]
an Other Unit circle radius 1 but Q at centers of it. has point P on the square corner. horizontal line thru Q will do angle λ + [π / 2] between P and Q and the line
P point equaling [ cos λ + cos[λ + [π / 2]] , sin λ + sin[λ + [π / 2]] ]
now Unit circle radius √2 with origin centered is will also has P. but angle λ + [π / 4]
P point equaling [ √2 cos[λ + [π / 4]] , √2 sin[λ + [π / 4]] ]
now and cos λ + cos[λ + [π / 2]] = √2 cos[λ + [π / 4]]
now and sin λ + sin[λ + [π / 2]] = √2 sin[λ + [π / 4]]
moonlune wrote
I tried a different approach for fun.
this system is just the real and imaginary part of the same complex:
which can be written:
QED.