The Identities

In trigonometry, there are six sum and difference identities. They are useful when the given angle in a trigonometry expression cannot be evaluated. The six sum and difference identities are given as:

$$\begin{align}& \text{1.) } sin(a + b) = sin\,a\:cos\,b + cos\,a\:sin\,b \\ \\ & \text{2.) } sin(a - b) = sin\,a\:cos\,b - cos\,a\:sin\,b \\ \\ & \text{3.) } cos(a + b) = cos\,a\:cos\,b - sin\,a\:sin\,b \\ \\ & \text{4.) } cos(a - b) = cos\,a\:cos\,b + sin\,a\:sin\,b \\ \\ & \text{5.) } tan(a + b) = \frac{tan\,a + tan\,b}{1 - tan\,a\:tan\,b} \\ \\ & \text{6.) } tan(a - b) = \frac{tan\,a - tan\,b}{1 + tan\,a\:tan\,b} \end{align}$$

Sum and Difference Identities Example Problem

Let's work through an example problem together to practice using the identities.

$$\begin{align}& \text{Evaluate the following three trigonometry expressions by} \\ & \text{using the sum and difference identities.} \\ \\ & \text{1.) } sin(\frac{\pi}{12}) \\ \\ & \text{2.) } cos(\frac{\pi}{12})\\ \\ & \text{3.) } tan(\frac{\pi}{12}) \\ \\ & \text{Solution:} \\ \\ & \text{To evaluate the angle } \frac{\pi}{12} \text{ without a calculator we can use } \frac{\pi}{4} - \frac{\pi}{6} \text{.} \\ \\ & \text{This means we use the difference identities where } a - b = \frac{\pi}{4} - \frac{\pi}{6} \text{.} \\ \\ & \text{Therefore, } a = \frac{\pi}{4} \text{ and } b = \frac{\pi}{6} \text{.} \\ \\ & \text{1.) } sin(\frac{\pi}{12}) = sin(\frac{\pi}{4}) \: cos(\frac{\pi}{6}) - cos(\frac{\pi}{4}) \: sin(\frac{\pi}{6}) \\ & \hspace{3ex} \text{The final answer is } \frac{\sqrt{6} - \sqrt{2}}{4} \text{ or } 0.259 \text{.} \\ \\ & \text{2.) } cos(\frac{\pi}{12}) = cos(\frac{\pi}{4}) \: cos(\frac{\pi}{6}) + sin(\frac{\pi}{4}) \: sin(\frac{\pi}{6}) \\ & \hspace{3ex} \text{The final answer is } \frac{\sqrt{6} + \sqrt{2}}{4} \text{ or } 0.966 \text{.} \\ \\ & \text{3.) } tan(\frac{\pi}{12}) = \frac{tan(\frac{\pi}{4}) - tan(\frac{\pi}{6})}{1 + tan(\frac{\pi}{4})\:tan(\frac{\pi}{6})} \\ & \hspace{3ex} \text{The final answer is } 2 - \sqrt{3} \text{ or } 0.268 \text{.} \end{align}$$