Question

prove that sin-cos+1/sin+cos-1=1/Sec-tan​

Answer 1

GIVEN:-

LHS=

( sinx-cosx+1) / (sinx+cosx-1)

TO PROVE:-

RHS =

secx-tanx. ( after correction)

SOLUTION:-

LHS=

(sinx-cosx+1) / (sinx+cosx-1)

multipliying by numerator in numerator and denominator both we have.

= [sinx-(cosx-1)]² / [ sin²x-(cosx-1)²]

sin²x+(cosx-1)²-2sinx(cosx-1)

= ------------------------------------------

sin²x-cos²x-1+2cosx

sin²x+cos²x+1-2cosx-2sinxcosx+2sinx

= -------------------------------------------------------

-2cos²x+2 cosx

2-2cosx-2sinxcosx+2sinx

= -----------------------------------------

-2cos²x +2cosx

2(1-cosx)-2sinx(1-cosx)

= -------------------------------------

2cosx(1-cosx)

2 (1-cosx) (1-sinx)

=. -------------------------

2(1-cosx) cosx

1-sinx

= ------------

cosx

= (1/cosx) - (sinx/cosx)

= secx-tanx

= RHS

HOPE THIS HELPS YOU