Difference of two Angles Identities

The difference of two angles are two trigonometric identities that express the cosine and sine of the difference of two angles in terms of their individual components. The two identities are given in the equations below:

Visual Proof

Difference of Two Angles Identites

The two identities can be visualized as a “proof-without-words” shown above.

Derive Difference of Two Angles Identities
Difference of Two Angles Identity Goal

To derive the sum of two angles identities a triangle can be drawn with a hypotenuse of length , an adjacent side of length and an opposite side of length . Naturally, the angle of and the angle of are drawn as well. To derive expressions for the dimensions of this right triangle, another right triangle with an angle is related to the first.

Steps
  1. Start by placing another right-triangle of hypotenuse and an angle of on top of the first right-triangle. The side lengths of this triangle is and .

    Derive Difference of Two Angles Step 1
  2. Then, the dimensions of the right triangle defined by the angle and the hypotenuse of length can be calculated. For clarity, the reflected triangle’s sides are labeled.

    Derive Difference of Two Angles Step 2 A

    Next we can observe that by finding the unknown lengths labeled below we have all the information needed to finish deriving the difference of two angles equations.

    Derive Difference of Two Angles Step 2 B

    Here are the equations for the difference of two angles so far:

  3. Observe that the right-triangle whose lengths we are interested in has an angle of and an hypotenuse of length .

    Derive Difference of Two Angles Step 3

    This gives us all the lengths of all the sides in question as shown in the image below:

    Difference of two Angles Identities

    Finally, we can substitue the lengths into the equations from step 2 to finish deriving the two identities.