Question

If secA-tanA=x show that x^2+1/x^2-1 =-cosecA

Answer 1

Answer:Given :  x  =  Sec A -  tan A ,

To show x2 + 1x2 − 1 = − Cosec A ,

We take LHS :

        x2 + 1 / x2 − 1 =

(Sec A − tan A)2 + 1 / (Sec − tan A)2  − 1

= Sec2 A + tan 2A − 2 Sec A tan A+ 1 / Sec2 A + tan 2A − 2 Sec A tan A  − 1

= Sec2 A + (tan 2A + 1) − 2 Sec A tan A / tan 2A +( Sec2 A  − 1) − 2 Sec A tan A

= Sec2 A + Sec2 A− 2 Sec A tan A / tan 2A +tan 2− 2 Sec A tan A  

[We know  Sec2− tan 2A =1]

= 2 Sec2 A − 2 Sec A tan A / 2 tan 2A − 2 Sec A tan A

=  2 Sec A  (Sec  A −tan A) / 2 tan A  (tan A −Sec  A )

=  − 2 Sec A   (tan A −Sec  A ) / 2 tan A  (tan A −Sec  A )

=  −  Sec A  /  tan A

=  −  Sec A  / Sin A /CosA

  [tan θ = Sin θ/Cos θ]

=  − Cosec A Sec A  / Sec A                                      

[Cosec θ = 1 / Sin θ , Sec θ =  1/ Cos θ]

= − Cosec A

So,

LHS =  RHS                                                   ( Hence proved )

Answer 2

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