you're getting sloppy ... sin, cos, sec, etc. mean nothing without the x.
also ... use the proper grouping symbols as shown below.
(1+sinx)/cosx + cosx/(1+sinx) = 2secx
common denominator for the left side is cosx(1+sinx) ...
(1+sinx)<sup>2</sup>/[cosx(1+sinx)] + cos<sup>2</sup>x/[cosx(1+sinx)] =
[(1+sinx)<sup>2</sup> + cos<sup>2</sup>x]/[cosx(1+sinx)] =
(1+2sinx+sin<sup>2</sup>x+cos<sup>2</sup>x)/[cosx(1+sinx)] =
(2+2sinx)/[cosx(1+sinx)] =
2(1+sinx)/[cosx(1+sinx)] =
2/cosx = 2secx