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    NationalHardTrigonometry10–12

    Proving a Trigonometric Identity

    Demonstrar uma Identidade Trigonométrica

    Show that 1−cos⁡2xsin⁡xcos⁡x=tan⁡x\frac{1 - \cos^{2} x}{\sin x \cos x} = \tan x for every xx for which both sides are defined, and state which values of xx must be excluded.

    Work on one side only and look for the Pythagorean identity hiding in the numerator; the exclusions come from the denominators, not from the algebra.

    Solution

    Step 1 of 5

    1. 1.Start from the left-hand side. By the Pythagorean identity sin⁡2x+cos⁡2x=1\sin^{2} x + \cos^{2} x = 1, the numerator is