A scratchpad for testing some maths markup, including the MITCH FORMULA, where f ( t ) = sin ( α + ω t ) {\displaystyle f(t)=\sin(\alpha +\omega t)} and A = f ( 0 ) , B = f ( T ) , C = f ( 2 T ) , D = f ( 3 T ) {\displaystyle A=f(0),B=f(T),C=f(2T),D=f(3T)} .
A D − B C B B − C A + C C − B D {\displaystyle {AD-BC \over BB-CA+CC-BD}}
= sin ( α ) sin ( α + 3 ω T ) − sin ( α + ω T ) sin ( α + 2 ω T ) ( e t c . ) {\displaystyle ={\sin(\alpha )\sin(\alpha +3\omega T)-\sin(\alpha +\omega T)\sin(\alpha +2\omega T) \over (etc.)}}
= cos ( 3 ω T ) − cos ( 2 α + 3 ω T ) − cos ( ω T ) + cos ( 2 α + 3 ω T ) cos ( 0 ) − cos ( 2 α + 2 ω T ) − cos ( 2 ω T ) + cos ( 2 α + 2 ω T ) + cos ( 0 ) − cos ( 2 α + 4 ω T ) − cos ( 2 ω T ) + cos ( 2 α + 4 ω T ) {\displaystyle ={\cos(3\omega T)-{\cancel {\cos(2\alpha +3\omega T)}}-\cos(\omega T)+{\cancel {\cos(2\alpha +3\omega T)}} \over \cos(0)-{\cancel {\cos(2\alpha +2\omega T)}}-\cos(2\omega T)+{\cancel {\cos(2\alpha +2\omega T)}}+\cos(0)-{\cancel {\cos(2\alpha +4\omega T)}}-\cos(2\omega T)+{\cancel {\cos(2\alpha +4\omega T)}}}}
= cos ( 3 ω T ) − cos ( ω T ) 2 ( 1 − cos ( 2 ω T ) ) {\displaystyle ={\cos(3\omega T)-\cos(\omega T) \over 2(1-\cos(2\omega T))}}
= − 2 sin ( 2 ω T ) sin ( ω T ) 4 sin 2 ( ω T ) = − sin 2 ω T 2 sin ω T = − 2 sin ω T cos ω T 2 sin ω T = − cos ω T {\displaystyle ={-2\sin(2\omega T)\sin(\omega T) \over 4\sin ^{2}(\omega T)}={-\sin 2\omega T \over 2\sin \omega T}={-2\sin \omega T\cos \omega T \over 2\sin \omega T}=-\cos \omega T}
and 
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