Sum and Difference Identities

How can the height of a mountain be measured? What about the distance from Earth to the sun? Like many seemingly impossible problems, we rely on mathematical formulas to find the answers. The trigonometric identities, commonly used in mathematical proofs, have had real-world applications for centuries, including their use in calculating long distances.

The trigonometric identities we will examine in this section can be traced to a Persian astronomer who lived around 950 AD, but the ancient Greeks discovered these same formulas much earlier and stated them in terms of chords. These are special equations or postulates, true for all values input to the equations, and with innumerable applications.

In this section, we will learn techniques that will enable us to solve problems such as the ones presented above. The formulas that follow will simplify many trigonometric expressions and equations. Keep in mind that, throughout this section, the term formula is used synonymously with the word identity.

Using the Sum and Difference Formulas for Cosine

Finding the exact value of the sine, cosine, or tangent of an angle is often easier if we can rewrite the given angle in terms of two angles that have known trigonometric values. We can use the special angles, which we can review in the unit circle shown below.

We will begin with the sum and difference formulas for cosine, so that we can find the cosine of a given angle if we can break it up into the sum or difference of two of the special angles. See the table below.

First, we will prove the difference formula for cosines. Let’s consider two points on the unit circle. Point $P$ is at an angle $\alpha$ from the positive $x$-axis with coordinates $(\cos\alpha,\sin\alpha)$ and point $Q$ is at an angle of $\beta$ from the positive $x$-axis with coordinates $(\cos\beta,\sin\beta)$. Note the measure of angle $POQ$ is $\alpha-\beta$.

Label two more points: $A$ at an angle of $(\alpha-\beta)$ from the positive $x$-axis with coordinates $(\cos(\alpha-\beta),\sin(\alpha-\beta))$; and point $B$ with coordinates $(1,0)$. Triangle $POQ$ is a rotation of triangle $AOB$ and thus the distance from $P$ to $Q$ is the same as the distance from $A$ to $B$.

We can find the distance from $P$ to $Q$ using the distance formula.

Then we apply the Pythagorean Identity and simplify.

Similarly, using the distance formula we can find the distance from $A$ to $B$.

Applying the Pythagorean Identity and simplifying we get:

Because the two distances are the same, we set them equal to each other and simplify.

Finally we subtract $2$ from both sides and divide both sides by $-2$.

Thus, we have the difference formula for cosine. We can use similar methods to derive the cosine of the sum of two angles.

Example. Using the formula for the cosine of the difference of two angles, find the exact value of $\cos\left(\tfrac{5\pi}{4}-\tfrac{\pi}{6}\right)$.

Solution. Use the formula for the cosine of the difference of two angles. We have

Example. Find the exact value of $\cos(75^\circ)$.

Solution. As $75^\circ=45^\circ+30^\circ$, we can evaluate $\cos(75^\circ)$ as $\cos(45^\circ+30^\circ)$. Thus,

Using the Sum and Difference Formulas for Sine

The sum and difference formulas for sine can be derived in the same manner as those for cosine, and they resemble the cosine formulas.

Example. Use the sum and difference identities to evaluate the difference of the angles and show that part ⓐ equals part ⓑ.

ⓐ $\sin(45^\circ-30^\circ)$ ⓑ $\sin(135^\circ-120^\circ)$

Solution. ⓐ Let’s begin by writing the formula and substitute the given angles.

Next, we need to find the values of the trigonometric expressions.

Now we can substitute these values into the equation and simplify.

ⓑ Again, we write the formula and substitute the given angles.

Next, we find the values of the trigonometric expressions.

Now we can substitute these values into the equation and simplify.

Example. Find the exact value of $\sin\left(\cos^{-1}\tfrac12+\sin^{-1}\tfrac35\right)$.

Solution. The pattern displayed in this problem is $\sin(\alpha+\beta)$. Let $\alpha=\cos^{-1}\tfrac12$ and $\beta=\sin^{-1}\tfrac35$. Then we can write

We will use the Pythagorean Identities to find $\sin\alpha$ and $\cos\beta$.

Using the sum formula for sine,

Using the Sum and Difference Formulas for Tangent

Finding exact values for the tangent of the sum or difference of two angles is a little more complicated, but again, it is a matter of recognizing the pattern.

Finding the sum of two angles formula for tangent involves taking the quotient of the sum formulas for sine and cosine and simplifying. Recall, $\tan x=\tfrac{\sin x}{\cos x},\ \cos x\ne0$.

Let’s derive the sum formula for tangent.

We can derive the difference formula for tangent in a similar way.

Example. Find the exact value of $\tan\left(\tfrac{\pi}{6}+\tfrac{\pi}{4}\right)$.

Solution. Let’s first write the sum formula for tangent and substitute the given angles into the formula.

Next, we determine the individual tangents within the formula:

So we have

Example. Given $\sin\alpha=\tfrac35,\ 0<\alpha<\tfrac{\pi}{2},\ \cos\beta=-\tfrac{5}{13},\ \pi<\beta<\tfrac{3\pi}{2}$, find

ⓐ $\sin(\alpha+\beta)$ ⓑ $\cos(\alpha+\beta)$ ⓒ $\tan(\alpha+\beta)$ ⓓ $\tan(\alpha-\beta)$

Solution. We can use the sum and difference formulas to identify the sum or difference of angles when the ratio of sine, cosine, or tangent is provided for each of the individual angles. To do so, we construct what is called a reference triangle to help find each component of the sum and difference formulas.

ⓐ To find $\sin(\alpha+\beta)$, we begin with $\sin\alpha=\tfrac35$ and $0<\alpha<\tfrac{\pi}{2}$. The side opposite $\alpha$ has length 3, the hypotenuse has length 5, and $\alpha$ is in the first quadrant. See the figure below. Using the Pythagorean Theorem, we can find the length of side $a$:

Since $\cos\beta=-\tfrac{5}{13}$ and $\pi<\beta<\tfrac{3\pi}{2}$, the side adjacent to $\beta$ is $-5$, the hypotenuse is 13, and $\beta$ is in the third quadrant. See the figure below. Again, using the Pythagorean Theorem, we have

Since $\beta$ is in the third quadrant, $a=-12$.

The next step is finding the cosine of $\alpha$ and the sine of $\beta$. The cosine of $\alpha$ is the adjacent side over the hypotenuse. We can find it from the triangle above: $\cos\alpha=\tfrac45$. We can also find the sine of $\beta$ from the triangle above, as opposite side over the hypotenuse: $\sin\beta=-\tfrac{12}{13}$. Now we are ready to evaluate $\sin(\alpha+\beta)$.

ⓑ We can find $\cos(\alpha+\beta)$ in a similar manner. We substitute the values according to the formula.

ⓒ For $\tan(\alpha+\beta)$, if $\sin\alpha=\tfrac35$ and $\cos\alpha=\tfrac45$, then

If $\sin\beta=-\tfrac{12}{13}$ and $\cos\beta=-\tfrac{5}{13}$, then

Then,

ⓓ To find $\tan(\alpha-\beta)$, we have the values we need. We can substitute them in and evaluate.

Analysis. A common mistake when addressing problems such as this one is that we may be tempted to think that $\alpha$ and $\beta$ are angles in the same triangle, which of course, they are not. Also note that

Using Sum and Difference Formulas for Cofunctions

Now that we can find the sine, cosine, and tangent functions for the sums and differences of angles, we can use them to do the same for their cofunctions. You may recall from Right Triangle Trigonometry that, if the sum of two positive angles is $\tfrac{\pi}{2}$, those two angles are complements, and the sum of the two acute angles in a right triangle is $\tfrac{\pi}{2}$, so they are also complements. In the figure below, notice that if one of the acute angles is labeled as $\theta$, then the other acute angle must be labeled $\left(\tfrac{\pi}{2}-\theta\right)$.

Notice also that $\sin\theta=\cos\left(\tfrac{\pi}{2}-\theta\right)$: opposite over hypotenuse. Thus, when two angles are complementary, we can say that the sine of $\theta$ equals the cofunction of the complement of $\theta$. Similarly, tangent and cotangent are cofunctions, and secant and cosecant are cofunctions.

From these relationships, the cofunction identities are formed.

Notice that the formulas in the table may also be justified algebraically using the sum and difference formulas. For example, using

we can write

Example. Write $\tan\tfrac{\pi}{9}$ in terms of its cofunction.

Solution. The cofunction of $\tan\theta=\cot\left(\tfrac{\pi}{2}-\theta\right)$. Thus,

Using the Sum and Difference Formulas to Verify Identities

Verifying an identity means demonstrating that the equation holds for all values of the variable. It helps to be very familiar with the identities or to have a list of them accessible while working the problems. Reviewing the general rules from Simplifying and Verifying Trigonometric Identities may help simplify the process of verifying an identity.

Example. Verify the identity $\sin(\alpha+\beta)+\sin(\alpha-\beta)=2\sin\alpha\cos\beta$.

Solution. We see that the left side of the equation includes the sines of the sum and the difference of angles.

We can rewrite each using the sum and difference formulas.

We see that the identity is verified.

Example. Verify the following identity.

Solution. We can begin by rewriting the numerator on the left side of the equation.

We see that the identity is verified. In many cases, verifying tangent identities can successfully be accomplished by writing the tangent in terms of sine and cosine.

Example. Let $L_1$ and $L_2$ denote two non-vertical intersecting lines, and let $\theta$ denote the acute angle between $L_1$ and $L_2$. See the figure below. Show that

where $m_1$ and $m_2$ are the slopes of $L_1$ and $L_2$ respectively. (Hint: Use the fact that $\tan\theta_1=m_1$ and $\tan\theta_2=m_2$.)

Solution. Using the difference formula for tangent, this problem does not seem as daunting as it might.

Example. For a climbing wall, a guy-wire $R$ is attached 47 feet high on a vertical pole. Added support is provided by another guy-wire $S$ attached 40 feet above ground on the same pole. If the wires are attached to the ground 50 feet from the pole, find the angle $\alpha$ between the wires. See the figure below.

Solution. Let’s first summarize the information we can gather from the diagram. As only the sides adjacent to the right angle are known, we can use the tangent function. Notice that $\tan\beta=\tfrac{47}{50}$, and $\tan(\beta-\alpha)=\tfrac{40}{50}=\tfrac45$. We can then use the difference formula for tangent.

Now, substituting the values we know into the formula, we have

Use the distributive property, and then simplify the functions.

Now we can calculate the angle in degrees.

Analysis. Occasionally, when an application appears that includes a right triangle, we may think that solving is a matter of applying the Pythagorean Theorem. That may be partially true, but it depends on what the problem is asking and what information is given.

Key equations

Key concepts

• The sum formula for cosines states that the cosine of the sum of two angles equals the product of the cosines of the angles minus the product of the sines of the angles. The difference formula for cosines states that the cosine of the difference of two angles equals the product of the cosines of the angles plus the product of the sines of the angles.
• The sum and difference formulas can be used to find the exact values of the sine, cosine, or tangent of an angle.
• The sum formula for sines states that the sine of the sum of two angles equals the product of the sine of the first angle and cosine of the second angle plus the product of the cosine of the first angle and the sine of the second angle. The difference formula for sines states that the sine of the difference of two angles equals the product of the sine of the first angle and cosine of the second angle minus the product of the cosine of the first angle and the sine of the second angle.
• The sum and difference formulas for sine and cosine can also be used for inverse trigonometric functions.
• The sum formula for tangent states that the tangent of the sum of two angles equals the sum of the tangents of the angles divided by 1 minus the product of the tangents of the angles. The difference formula for tangent states that the tangent of the difference of two angles equals the difference of the tangents of the angles divided by 1 plus the product of the tangents of the angles.
• The Pythagorean Theorem along with the sum and difference formulas can be used to find multiple sums and differences of angles.
• The cofunction identities apply to complementary angles and pairs of reciprocal functions.
• Sum and difference formulas are useful in verifying identities.
• Application problems are often easier to solve by using sum and difference formulas.

Practice

Use sum and difference formulas for cosine.

Use sum and difference formulas for sine.

Use sum and difference formulas for tangent.

Use sum and difference formulas for cofunctions.

Use sum and difference formulas to verify identities.

This section is adapted from Precalculus 2e, Section 7.2: Sum and Difference Identities by Jay Abramson and OpenStax, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: omitted the decorative photograph of Denali (Figure 1). Recreated all seven instructional figures as accessible spec-first SVGs: the unit circle of special angles (Figure 2), reusing the coordinate-labeled unit circle already authored for Section 5.2 with its degree/radian ray labels omitted, since the (cosine, sine) coordinates alone carry this figure’s reason for being shown here (deriving the sum and difference formulas), and every special angle’s degree and radian measure is transcribed in full in Sections 5.2 and 5.3; the difference-formula proof diagram (Figure 3), rebuilt with representative angles $\alpha=140^\circ$, $\beta=20^\circ$ standing in for the source’s generic $\alpha,\beta$, since the proof depends only on the two points’ being distinct and in general position, not on their specific measures; the two reference-triangle figures for Example 6 (Figures 4–5); the cofunction right triangle (Figure 6); the two-line application diagram (Figure 7); and the guy-wire diagram (Figure 8), with the source’s separate dashed horizontal reference ray for angles $\alpha$ and $\beta$ replaced by the solid $50$-foot ground segment itself, which is both the same reference direction and the segment the problem already measures. Every retained Try It became a real fillin or multiplechoice component. Two Try Its whose printed subject is itself a single trigonometric application of a cofunction- or supplement-related argument — “write $\sin\tfrac{\pi}{7}$ in terms of its cofunction” and “verify $\tan(\pi-\theta)=-\tan\theta$” — were authored as multiplechoice rather than fillin: since both the printed subject and the keyed answer are exactly one trigonometric application of the variable, the grader’s single-trig-function token cannot refuse a learner who simply retypes the prompt, so the response mode changed instead of the token; the corresponding cofunction Practice items ($\sec\left(\tfrac{\pi}{2}-\theta\right)$, $\tan\left(\tfrac{\pi}{2}-x\right)$) received the same adaptation. One Extension “prove or disprove” item and one Graphical “same or different” item, adapted into the closing Practice block’s verify-identities group, were changed from free-response proofs to multiplechoice for the same reason a proof has no gradable free-response answer; their keyed alternative is the source’s own printed solution (“True” / “They are the same”). Omitted the “Access these online resources” media links. The two-line application example’s stated goal is corrected from the source’s $\tfrac{m_2-m_1}{1+m_1m_2}$ to the $\tfrac{m_1-m_2}{1+m_1m_2}$ its own solution derives and its own figure supports, with a visible Source note beside the statement (the source’s stated goal is the negative of what its solution proves). The five source Try Its (three fillin, two multiplechoice) were transcribed as in-page practice following their examples; ten further end-of-section exercises — two per objective, covering exact-value evaluation, rewriting in terms of $\sin x$ and $\cos x$, a tangent-difference simplification, cofunction simplification, and identity/graph-comparison recognition — were adapted into the closing Practice block, one group per objective.