To prove that sinQ-cosQ+1/sinQ+cosQ-1 =1/secQ-tanQ
Consider LHS
LHS => SinQ-cosQ+1/sinQ+cosQ+1
=> (sinQ/cosQ-cosQ/cosQ+1/cosQ)/(sinQ/cosQ+cosQ/cosQ-1/cosQ) (divide nominator and denominate by cosQ)
=> (tanQ-1+secQ)/(tanQ+1-secQ)
=> (tanQ+secQ-1)/(tanQ-secQ+1)
=> (tanQ+secQ-1)/(tanQ-secQ+(sec²Q-tan²Q) (since sec²Q-tan²Q=1)
=> (tanQ+secQ-1)/(tanQ-secQ+[(secQ-tanQ)(secQ+tanQ)]
=> (tanQ+secQ-1)/-(secQ-tanQ)+[(secQ-tanQ)(secQ+tanQ)]
=>(tanQ+secQ-1)/(secQ-tanQ)[(secQ+tanQ)-1]
=> 1/secQ-tanQ (since (tanQ+secQ-1)/[(secQ+tanQ)-1]=1)
=> RHS
=> Hence Proved
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