How to Prove an Equality Using Pythagorean Identities

When asked to prove an identity, if you see a negative of a variable inside a trig function, you automatically use an even/odd identity. You first replace all trig functions with a negative variable inside the parentheses with the correct trig function using a positive variable by making use of the even/odd identities. Then you simplify the trig expression to make one side look like the other side. Here’s just one example of how this works.

With the following steps, prove this identity:

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  1. Working with the left side, replace all negative angles and their trig functions with the even/odd identity that matches.

    image1.png
  2. Simplify the new expression.

    Because the right side doesn’t have any fractions in it, eliminating the fractions from the left side is an excellent place to start. In order to subtract fractions, you first must find a common denominator. However, before doing that, observe that the first fraction can be split up into the sum of two fractions, as can the second fraction. By doing this step first, certain terms simplify and make your job much easier when the time comes to work with the fractions.

    Therefore, you get

    image2.png

    which quickly simplifies to

    image3.png

    Now you must find a common denominator. For this example, the common denominator is

    image4.png

    Multiplying the first term by

    image5.png

    and the second term by

    image6.png

    gives you

    image7.png

    You can rewrite this equation as

    image8.png

    Here is a Pythagorean identity in its finest form!

    image9.png

    is the most frequently used of the Pythagorean identities. This equation then simplifies to

    image10.png

    Using the reciprocal identities, you get

    image11.png
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