• We will extend our knowledge of the fundamental trigonometric identities to include the compound angle identities shown here.

  \(\sin\left(A + B\right)\)

  \(=\sin(A)\cos(B) + \cos(A)\sin(B)\)

  \(\sin\left(A - B\right)\)

  \(=\sin(A)\cos(B) - \cos(A)\sin(B)\)

  \(\cos\left(A + B\right)\)

  \(=\cos(A)\cos(B) - \sin(A)\sin(B)\)

  \(\cos\left(A - B\right)\)

  \(=\cos(A)\cos(B) + \sin(A)\sin(B)\)

  \(\tan\left(A + B\right)\)

  \(=\dfrac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)}\)

  \(\tan\left(A - B\right)\)

  \(=\dfrac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)}\)

These identities involve the sum and difference of two angles and are often referred to as formulas as they provide a means of

• simplifying trigonometric expressions and proving other identities,
• determining exact values for angles related to the acute angles \(\dfrac{\pi}{12}\) or \(\dfrac{5\pi}{12}\) \(\left(15^{\circ} \right.\) or \(\left. 75^{\circ} \right)\), and
• solving certain trigonometric equations.

We will begin by deriving the formula for \(\cos\left(A + B \right)\) using the unit circle.

Consider the two points \(P\) and \(Q\) on the unit circle, where \(P\) is defined by \(\left( \cos(\theta) , \sin(\theta) \right)\) for some angle \(\theta\), \(\theta \gt 0\) and \(Q\) is given by \(\left(\cos(-\beta) , \sin(-\beta)\right)\) for some angle \(\beta\), \(\beta \gt 0\). The coordinates of \(Q\) can be simplified to \(\left(\cos(\beta) , -\sin(\beta)\right)\).

The measure of \(\angle POQ\), with \(O\) at the origin, is \(\theta + \beta\).

The length of the line segment \(PQ\) can be found using

Thus,

Now rotate the points \(P\) and \(Q\) counterclockwise about the origin by angle of \(\beta\) to obtain \({\color{NavyBlue}Q'(1,0)}\) and \({\color{NavyBlue}P'~(\cos(\theta + \beta) , \sin(\theta + \beta))}\) on the unit circle, as shown in the diagram.

As a condition of the rotation, \(\angle P'OQ' = \theta + \beta\) and \(\vert P'Q' \rvert = \lvert PQ \rvert\).

The length of \(P'Q'\) is given by

Since \(\lvert P'Q' \rvert = \lvert PQ \rvert\), we have