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English: as it is easy to proof things using imaginary numbers starting with derivatives of sin(x) and cos(x) and tan(x) and x to power of w. another line shows how easy it is to prove pithagoras equation using imaginary numbers. one line below is proof for moivre's equation. subsequent thing is proof for square angle addition to cos function. one before end is proof for subtraction of cosines on imaginary numbers. last thing is proof for additon of combination of euler Beta function