Can anyone prove, and show step by step, that

1 + cosx + sinx = secx + tanx
1 + cosx - sinx

Because I am absolutely stumped.

Same advice as with your other recent post - convert the secant and tangent functions to their sine and cosibne equivalents, and simplfy. The only trick you will have to use is to recognize that 1-sin^2x = cos^2x.