prove that sinQ-cosQ+1/sinQ+cosQ-1 =1/secQ-tanQ

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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