Prove sinx/secx+tanx-1 + cosx/cosec x+cotx-1 = 1

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We have to prove that: sin x /(sec x + tan x - 1) + cos x /(cosec x + cot x - 1) = 1

sin x /(sec x + tan x - 1) + cos x /(cosec x + cot x - 1)

=> sin x /(1/cos x + sin x/cos x - 1) + cos x /(1/sin x + cos x/sin x - 1)

=> sin x /(1/cos x + sin x/cos x - cos x/cos x) + cos x /(1/sin x + cos x/sin x - sin x/sin x)

=> sin x * cos x/(1+ sin x - cos x) + cos x*sin x /(1 + cos x - sin x)

=> sin x * cos x[1/(1+ sin x - cos x) + 1/(1 + cos x - sin x)]

=> sin x * cos x[(1 + cos x - sin x) + (1+ sin x - cos x)/(1 + cos x - sin x)(1+ sin x - cos x)]

=> sin x * cos x[(1 + cos x - sin x + 1+ sin x - cos x)/(1^2 - (sin x+ cos x)^2]

=> sin x * cos x[(1 + 1)/(1 - (sin x)^2 - (cos x)^2 + 2*sin x*cos x]

=> sin x * cos x[2/(1 - 1 + 2*sin x*cos x]

=> sin x * cos x[2/2*sin x*cos x]

=> sin x * cos x/sin x*cos x

=> 1

This proves that sin x /(sec x + tan x - 1) + cos x /(cosec x + cot x - 1) = 1

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