Question

√1+cos x/√1-cos x=Cosec x + cot x
Prove that

Answer 1

Multiply numerator and denominator by √{1+cos} to get (1+cos)/sin = cosec + cot

Answer 2

LHS = 1/sin x(1/cos x-1) - cos x/sinx (1-cos x)

=1-cos x/sin xcos x -( 1-cosx) cosx/sinx

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

=(1-cosx/sinx)sin*2x/cosx

=(1-cosx)sinx/cosx

=sinx/cosx -sinx

=tanx-sinx=RHS

HENCE LHS=RHS

proved