Lesson 9.2: Sum and Difference Identities

New Concept

These powerful identities allow us to find exact trigonometric values for angles that are a sum or difference of special angles, like finding $\operatorname{cos}(75^\circ)$ by using $45^\circ$ and $30^\circ$. They are key for simplifying and verifying complex expressions.

What’s next

Next, you'll master the sum and difference formulas for sine, cosine, and tangent through a series of interactive examples and practice cards.

Property

These formulas can be used to calculate the cosine of sums and differences of angles.

To find the cosine of the difference or sum between two angles, write the appropriate formula, substitute the values of the given angles, and simplify.

Examples

• Find the exact value of $\operatorname{cos}(15^\circ)$. We can write this as $\operatorname{cos}(45^\circ - 30^\circ)$, which equals $\operatorname{cos}(45^\circ)\operatorname{cos}(30^\circ) + \operatorname{sin}(45^\circ)\operatorname{sin}(30^\circ) = (\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) + (\frac{\sqrt{2}}{2})(\frac{1}{2}) = \frac{\sqrt{6}+\sqrt{2}}{4}$.

• Find the exact value of $\operatorname{cos}(\frac{7\pi}{12})$. We use $\frac{7\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4}$. So, $\operatorname{cos}(\frac{\pi}{3} + \frac{\pi}{4}) = \operatorname{cos}(\frac{\pi}{3})\operatorname{cos}(\frac{\pi}{4}) - \operatorname{sin}(\frac{\pi}{3})\operatorname{sin}(\frac{\pi}{4}) = (\frac{1}{2})(\frac{\sqrt{2}}{2}) - (\frac{\sqrt{3}}{2})(\frac{\sqrt{2}}{2}) = \frac{\sqrt{2}-\sqrt{6}}{4}$.

Property

These formulas can be used to calculate the sines of sums and differences of angles.

To find the sine of the difference or sum between two angles, write the appropriate formula, substitute the values of the given angles, and simplify.

Examples

• Find the exact value of $\operatorname{sin}(105^\circ)$. We use $105^\circ = 60^\circ + 45^\circ$. So, $\operatorname{sin}(60^\circ + 45^\circ) = \operatorname{sin}(60^\circ)\operatorname{cos}(45^\circ) + \operatorname{cos}(60^\circ)\operatorname{sin}(45^\circ) = (\frac{\sqrt{3}}{2})(\frac{\sqrt{2}}{2}) + (\frac{1}{2})(\frac{\sqrt{2}}{2}) = \frac{\sqrt{6}+\sqrt{2}}{4}$.

• Find the exact value of $\operatorname{sin}(\frac{\pi}{12})$. We use $\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$. So, $\operatorname{sin}(\frac{\pi}{3} - \frac{\pi}{4}) = \operatorname{sin}(\frac{\pi}{3})\operatorname{cos}(\frac{\pi}{4}) - \operatorname{cos}(\frac{\pi}{3})\operatorname{sin}(\frac{\pi}{4}) = (\frac{\sqrt{3}}{2})(\frac{\sqrt{2}}{2}) - (\frac{1}{2})(\frac{\sqrt{2}}{2}) = \frac{\sqrt{6}-\sqrt{2}}{4}$.

Property

The sum and difference formulas for tangent are:

To find the tangent of the sum or difference of angles, write the appropriate formula, substitute the given angles, and simplify.

Examples

• Find the exact value of $\operatorname{tan}(75^\circ)$. As $75^\circ = 45^\circ + 30^\circ$, we have $\operatorname{tan}(45^\circ+30^\circ) = \frac{\operatorname{tan}(45^\circ)+\operatorname{tan}(30^\circ)}{1-\operatorname{tan}(45^\circ)\operatorname{tan}(30^\circ)} = \frac{1+\frac{1}{\sqrt{3}}}{1-1 \cdot \frac{1}{\sqrt{3}}} = \frac{\sqrt{3}+1}{\sqrt{3}-1} = 2+\sqrt{3}$.

• Find the exact value of $\operatorname{tan}(\frac{\pi}{12})$. As $\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$, we have $\operatorname{tan}(\frac{\pi}{3}-\frac{\pi}{4}) = \frac{\operatorname{tan}(\frac{\pi}{3})-\operatorname{tan}(\frac{\pi}{4})}{1+\operatorname{tan}(\frac{\pi}{3})\operatorname{tan}(\frac{\pi}{4})} = \frac{\sqrt{3}-1}{1+\sqrt{3}} = 2-\sqrt{3}$.

Property

The cofunction identities are summarized in the table below.
| Identity | Identity |
|---|---|
| $\operatorname{sin} \theta = \operatorname{cos}(\frac{\pi}{2} - \theta)$ | $\operatorname{cos} \theta = \operatorname{sin}(\frac{\pi}{2} - \theta)$ |
| $\operatorname{tan} \theta = \operatorname{cot}(\frac{\pi}{2} - \theta)$ | $\operatorname{cot} \theta = \operatorname{tan}(\frac{\pi}{2} - \theta)$ |
| $\operatorname{sec} \theta = \operatorname{csc}(\frac{\pi}{2} - \theta)$ | $\operatorname{csc} \theta = \operatorname{sec}(\frac{\pi}{2} - \theta)$ |

Examples

• Write $\operatorname{sin}(25^\circ)$ in terms of its cofunction. Since sine and cosine are cofunctions, $\operatorname{sin}(25^\circ) = \operatorname{cos}(90^\circ - 25^\circ) = \operatorname{cos}(65^\circ)$.

• Write $\operatorname{cot}(\frac{2\pi}{7})$ in terms of its cofunction. The cofunction of cotangent is tangent, so $\operatorname{cot}(\frac{2\pi}{7}) = \operatorname{tan}(\frac{\pi}{2} - \frac{2\pi}{7}) = \operatorname{tan}(\frac{3\pi}{14})$.

Property

To verify an identity, follow these steps:

1. Begin with the expression on the side of the equal sign that appears most complex.
2. Look for opportunities to use the sum and difference formulas.
3. Rewrite sums or differences of quotients as single quotients.
4. If the process becomes cumbersome, rewrite the expression in terms of sines and cosines.

Examples

• Verify the identity $\operatorname{cos}(\alpha + \beta) + \operatorname{cos}(\alpha - \beta) = 2 \operatorname{cos} \alpha \operatorname{cos} \beta$. Expand the left side: $(\operatorname{cos} \alpha \operatorname{cos} \beta - \operatorname{sin} \alpha \operatorname{sin} \beta) + (\operatorname{cos} \alpha \operatorname{cos} \beta + \operatorname{sin} \alpha \operatorname{sin} \beta) = 2 \operatorname{cos} \alpha \operatorname{cos} \beta$.

• Verify $\frac{\operatorname{cos}(\alpha - \beta)}{\operatorname{sin} \alpha \operatorname{cos} \beta} = \operatorname{cot} \alpha + \operatorname{tan} \beta$. Expand the numerator: $\frac{\operatorname{cos} \alpha \operatorname{cos} \beta + \operatorname{sin} \alpha \operatorname{sin} \beta}{\operatorname{sin} \alpha \operatorname{cos} \beta} = \frac{\operatorname{cos} \alpha \operatorname{cos} \beta}{\operatorname{sin} \alpha \operatorname{cos} \beta} + \frac{\operatorname{sin} \alpha \operatorname{sin} \beta}{\operatorname{sin} \alpha \operatorname{cos} \beta} = \operatorname{cot} \alpha + \operatorname{tan} \beta$.