Trigonometric identity problems require you to substitute values for the expressions on the left with equivalent values until the expression on the right is produced. This sometimes requires some trial and error before the correct expression is produced (don't get discouraged if you don't get the desired result on the first try).
secx - tanxsinx Given
1/cosx - (sinx/cosx) * sinx Subsitute 1/cosx for secx and sinx/cosx for tanx
1/cosx - sin2x/cosx Combine the sinx in the numerator with the original sinx in the second term
(1-sin2x)/cosx Combine the two terms over the same denominator (both were cosx)
cos2x/cos Substitute cos2x for 1 - sin2x (Pythagorean identity)
cosx Simplify the fraction
1/secx Substitute 1/secx for cosx