Question

Prove the following ::
tan x / 1-cot x + cot x / 1 - tan x = 1+ tanx +cot x = 1 + sec x cosec x

Answer 1

Refer to the file attached below.
Hope you'll get my handwriting.

Answer 2

Hey there!

LHS=tanx1−cotx+cotx1−tanx

=tan2xtanx(1−cotx)+cotx1−tanx

=cotx1−tanx+tan2xtanx−tanxcotx

=cotx1−tanx−tan2x1−tanx

=1tanx−tan2x1−tanx

=1−tan3xtanx(1−tanx)

=(1−tanx)(1+tanx+tan2x)tanx(1−tanx)

=tanx+sec2xtanx

=tanxtanx+sec2xtanx

=1+secx⋅secx⋅(cosxsinx)

=1+secx⋅(1sinx)

=1+secx⋅cscx=RHS

Hope it Helps You!