Sum and Difference Identities

Calculus Readiness · Axiom Academy

LESSON Sum and Difference Identities Break , , and into single-angle pieces — and read off exact values like . A quick test kills the naive guess. Take , so . Then , but . The guess is off by a full 2 in a single example. Here is the rule that actually works. Watch the animation: the true height gets rebuilt as a piece stacked on a piece. The two stack up to exactly the whole height. Three pairs — sine, cosine, tangent — each with a + and a - version. The sign pattern is the whole game, so the legend below pairs each with the one rule worth memorizing. The animation reads the tangent rule geometrically: a ray at A and a ray at B , with the difference angle A-B opening between them. Its slope is — exactly what the formula computes from the two separate slopes. uses + ; uses - . Same sign as inside. uses - ; uses + . Opposite of inside. Sine cosine, then cosine sine. Two different functions. Cosine cosine, then sine sine. Each with itself. Put B=A into the sine sum: . The double-angle formulas are just these six identities with both angles equal. Now the payoff. We cannot read off the unit circle directly — but , and both of those we know exactly. Apply the cosine difference rule (sign flips, so it becomes a + ): The animation builds the right-hand side one term at a time — first, then — collapsing to . The radius then swings to so you can see that value is the horizontal coordinate . Substitute the exact values , , to land the result:

This is the written version of the interactive lesson above. See the full Calculus Readiness course.