How to Prove Trigonometric Identities When You Start Off with Fractions

When the trig expression you're given begins with fractions, most of the time you have to add (or subtract) them to get things to simplify. Here's one example of a proof where doing just that gets the ball rolling. Say you have to find the lowest common denominator (LCD) to add the two fractions in order to simplify this expression:

image0.png

With that as the beginning step, follow along:

  1. In order to add these fractions, you must find the LCD of the two fractions.

    The least common denominator is

    image1.png

    so multiply the first term by

    image2.png

    and multiply the second term by

    image3.png

    You get

    image4.png
  2. Multiply or distribute in the numerators of the fractions.

    image5.png
  3. Add the two fractions.

    image6.png
  4. Look for any trig identities and substitute.

    You can rewrite the numerator as

    image7.png

    which is equal to

    image8.png

    because cos2 t + sin2 t = 1 (a Pythagorean identity).

  5. Cancel or reduce the fraction.

    After the top and the bottom are completely factored, you can cancel terms:

    image9.png
  6. Change any reciprocal trig functions.

    image10.png
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