I would work out the left side, since that side is the less simplified part.
You know that tanx = sinx / cosx
[1 / (sinx / cosx)] + (sinx / cosx) =
(cosx / sinx) + (sinx / cosx)
Now we look for the common denominator. The common denominator is sinx cosx.
(cos2 x + sin2 x ) / (sinx cosx)
The numerator part is Pythagorean theorem identity that is equal to 1.
1 / (sinx cosx)