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secx-tanxsinx=1/secx

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2 Answers

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