By Mary Jane Sterling

You need to become more familiar with the possibilities for rewriting trigonometric expressions. A trig identity is really an equivalent expression or form of a function that you can use in place of the original. The equivalent format may make factoring easier, solving an application possible, and (later) performing an operation in calculus more manageable.

The trigonometric identities are divided into many different classifications. These groupings help you remember the identities and make determining which identity to use in a particular substitution easier.

In a classic trig identity problem, you try to make one side of the equation match the other side. The best way to do so is to work on just one side — the left or the right — but sometimes you need to work on both sides to see just how to work the problem to the end.

In pre-calculus, you’ll work with the basic trigonometric identities in the following ways:

  • Determining which trig functions are reciprocals of one another

  • Creating Pythagorean identities from a right triangle whose hypotenuse measures 1 unit

  • Determining the sign of identities whose angle measure is negated

  • Matching up trig functions and their co-functions

  • Using the periods of functions in identities

  • Making the most of selected substitutions into identities

  • Working on only one side of the identity

  • Figuring out where to go with an identity by working both sides at once

Don’t let common mistakes trip you up; keep in mind that when working on trigonometric identities, some challenges will include the following:

  • Keeping track of where the 1 goes in the Pythagorean identities

  • Remembering the middle term when squaring binomials involving trig functions

  • Correctly rewriting Pythagorean identities when solving for a squared term

  • Recognizing the exponent notation

Practice problems

  1. Prove the trig identity. Indicate your first identity substitution:

    image0.jpg

    Answer: use the reciprocal identity

    Because each term contains a function and its reciprocal, using reciprocal identities will simplify the terms quickly.

    First, replace

    image1.jpg

    with its reciprocal identity,

    image2.jpg

    and

    image3.jpg

    with its reciprocal identity:

    image4.jpg

    Then simplify the complex fractions.

    image5.jpg

    Finally, replace

    image6.jpg

    with 1, using the Pythagorean identity: 1 = 1

  2. Determine the missing term or factor in the identity by changing all functions to those using sine or cosine:

    image7.jpg

    Answer: 1

    Use the reciprocal identity to replace csc2x and use the ratio identity to replace tan2x:

    image8.jpg

    Distribute

    image9.jpg

    simplify, and then combine the two terms:

    image10.jpg

    Rewrite the Pythagorean identity, sin2x + cos2x = 1, by subtracting sin2x from each side to get cos2x = 1 – sin2x. Replace the numerator of the fraction in the identity with cos2x:

    image11.jpg

    The missing term is 1.