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How to Apply the Sum and Difference Formulas for Cosine to Trig Proofs

You can use the sum and difference formulas for cosine to prove trig identities. When working with sum and difference formulas for cosines, you're simply plugging in given values for variables. Just make sure you use the correct formula based on the information you're given in the question.

Here are the sum and difference formulas for cosines:


For example, to prove


follow these steps:

  1. Outline the given information.

    You start with

  2. Look for sum and/or difference identities for cosine.

    In this case, the left side of the equation is the difference formula for cosine. Therefore, you can break up the first term by using the difference formula for cosines:

  3. Refer to the unit circle and substitute all the information you know.

    The whole unit circle
    The whole unit circle

    Using the unit circle, the equation simplifies to


    which equals sin x on the left-hand side. Your equation now says sin x = sin x. Ta-dah!

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