(cos x ÷ 1- sin x) - tan x= 1÷ cosx

Question

Prove the identity

Answer 1

(cos x ÷ 1- sin x) - tan x= 1÷ cosx

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

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

lhs= cos^2x/{cosx{1 - sinx)} - {sinx(1 - sinx)}/{cosx(1 - sinx)}

lhs  = [cos^2x - sinx + sin^2x] / {cosx(1 - sinx)}

lhs  = [1 - sinx] / {cosx(1 - sinx)}

lhs = 1/cosx = rhs

LHS = RHS

Answer 2

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

(cosx+cosxsinx)/(1-sin^2(x))-sinx/cosx=

(cosx+cosxsinx)/cos^2(x)-sinx/cosx=

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