How to Prove Trigonometric Identities When You Start Off with Fractions

By Yang Kuang, Elleyne Kase

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:

A trigonometric function with fractions.

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

    The lowest common denominator in a function.

    so multiply the first term by

    Cosine over cosine.

    and multiply the second term by

    1 plus sine divided by one plus sine.

    You get

    Two trigonometric expression with a common denominator.

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

    Multiplying or distributing the numerators of a fraction.

  3. Add the two fractions.

    Adding two fractions with a common denominator

  4. Look for any trig identities and substitute.

    You can rewrite the numerator as

    A Trigonometric expression with a fraction.

    which is equal to

    Replacing any trig identities in an expression.

    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:

    Factoring the components of a fraction.

  6. Change any reciprocal trig functions.

    Final result of trigonometric expression.