Mathematics, five units · Grade 12

How do you prove a trigonometric identity?

Work on one side only and transform it until it becomes the other. Do not move terms across, because that assumes the equality you are trying to establish. The method that almost always works is to write everything in terms of sine and cosine and then simplify using the fundamental identity.

Learning objectives

What an identity claims

An identity is not an equation to solve. It claims that two expressions are equal for every value of the variable, so there is nothing to find — there is something to demonstrate.

That difference decides the method. In an equation you may do the same thing to both sides, because you are assuming the equality holds and narrowing down where. In an identity you may not, because the equality is exactly what is in question.

So the rule is simple and absolute: pick one side, usually the more complicated one, and transform it by legitimate steps until it reads as the other.

The method that almost always works

Rewrite every function in terms of sine and cosine. Tangent becomes sine over cosine, and the rest follow. This alone turns most identities into ordinary fraction algebra.

Then look for the fundamental identity — sine squared plus cosine squared equals one — in either direction. Replacing a one by that sum is as legitimate a move as replacing the sum by one, and the second direction is the one people forget.

Finally, combine fractions over a common denominator and cancel. If nothing cancels, try the other side instead; some identities are far shorter from one end than from the other.

What you need to know by heart

The fundamental identity, the definition of tangent, and the double-angle formulas for sine and cosine. Almost every identity at this level is built from those.

Knowing the double-angle formula for cosine in all three of its forms is worth the effort, because choosing the right form is often the entire proof.

Worked examples

  1. Prove that tangent times cosine equals sine

    1. Start from the left side, which is the more complicated one
    2. Write the tangent as sine over cosine
    3. That gives sine over cosine, times cosine
    4. The cosine cancels

    Answer: The left side equals sine, which is the right side

  2. Prove that one minus sine squared equals cosine squared

    1. Start from the left side
    2. By the fundamental identity, one equals sine squared plus cosine squared
    3. Substitute: sine squared plus cosine squared, minus sine squared
    4. The sine squared terms cancel

    Answer: The left side equals cosine squared, as required

  3. Prove that sine over tangent equals cosine

    1. Start from the left side
    2. Write the tangent as sine over cosine
    3. Dividing by a fraction means multiplying by its reciprocal: sine times cosine over sine
    4. The sine cancels

    Answer: The left side equals cosine

Common mistakes

Moving terms across the equals sign
That assumes the identity is true, which is what you were asked to show. The proof will not be accepted however correct the algebra is afterwards.
Only ever replacing the sum of squares by one
The fundamental identity works in both directions, and replacing a one by sine squared plus cosine squared is often the step that unlocks the proof. Using it in one direction only leaves half the tool unused.
Cancelling across a sum
You may cancel a common factor, never a term inside a sum. Cancelling a sine from a numerator that is a sine plus something else changes the expression into a different one.

What to remember

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