secx-tanx=cosx/1+sinx

It's almost always best to start by subbing the basic sin and cosine terms in for the secant and tangent terms. Then multiply both sides by the denominator on the right (1 + sinx) and see what you get.

If the rhs is cosx/(1+sinx) then yes: rewrite lhs using cos and sin, massage and use 1-sinx*sinx = (1+sinx)(1-sinx) = cosx*cosx, massage again et voilà.