Prove:
sinA/(secA + tanA - 1) + cosA/(cosecA + cotA - 1) = 1
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sina/seca+tana-1+cosa/coseca+cota-1
=sina/(1/cosa+sina/cosa-1)+cosa/(1/sina+cosa/sina-1)
=sina/{(1+sina-cosa)/cosa}+cosa/{(1+cosa-sina)/sina}
=sinacosa/(1+sina-cosa)+sinacosa/(1+cosa-sina)
=sinacosa[(1+cosa-sina+1+sina-cosa)/(1+sina-cosa)(1+cosa-sina)]
=2sinacosa/(1+sina-cosa+cosa+sinacosa-cos²a-sina-sin²a+sinacosa)
=2sinacosa/{1+2sinacosa-(sin²a+cos²a)}
=2sinacosa/(1+2sinacosa-1)
=2sinacosa/2sinacosa
=1
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