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How to Prove an Equality by Using Periodicity Identities

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2016-03-26 15:10:44
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Pre-Calculus All-in-One For Dummies
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Using the periodicity identities comes in handy when you need to prove an equality that includes the expression (x + 2pi) or the addition (or subtraction) of the period. For example, to prove

image0.png

follow these steps:

  1. Replace all trig functions with the appropriate periodicity identity.

    You're left with (sec x – tan x)(csc x + 1).

  2. Simplify the new expression.

    For this example, the best place to start is to FOIL:

    image1.png

    Now convert all terms to sines and cosines to get

    image2.png

    Then find a common denominator and add the fractions:

    image3.png
  3. Apply any other applicable identities.

    You have a Pythagorean identity in the form of 1 – sin2x, so replace it with cos2x. Cancel one of the cosines in the numerator (because it's squared) with the cosine in the denominator to get

    image4.png

    Finally, this equation simplifies to cot x = cot x.

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