Show that ((1+tanx-secx)/(secx+tanx-1))= (1+secx-tanx)/(secx+tanx+1)?

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Answer 1 · 2018-06-27T08:54:11

Yes

Let

#u=1+secx+tanx# , #w=tanx-secx#

#(1+w)/(u-2)=(1-w)/u#

#cancelu+uw=cancelu-uw+2w-2#

#uw=w-1#

Substitute

#(1+secx+tanx)(tanx-secx)=tanx-secx-1#

#canceltanxcancel(-secx)cancel(+secxtanx)-sec^2x+tan^2xcancel(-secxtanx)=canceltanxcancel(-secx)-1#

#sec^2x-tan^2x=1#

#:.#

Answer 2 · 2018-06-27T14:11:49

We know

#sec^2x-tan^2x=1#

#=>(secx-tanx)(secx+tanx)=1#

#=>(secx+tanx)/1=1/(secx-tanx)#

By componendo and dividendo

#=>(secx+tanx+1)/(secx+tanx-1)=(1+secx-tanx)/(1-secx+tanx)#

#=>(1+tanx-secx)/(secx+tanx-1)=(1+secx-tanx)/(secx+tanx+1)#