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 Al-Khwarizmi, 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 in Figure 2.

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 Table 1.

Table 1

Sum formula for cosine	[latex]cos[/latex] ([latex]α[/latex] + [latex]β[/latex]) = [latex]cos α[/latex] [latex]cos β[/latex] − [latex]sin α[/latex] [latex]sin β[/latex]
Difference formula for cosine	[latex]cos[/latex] ([latex]α[/latex] − [latex]β[/latex]) = [latex]cos α[/latex] [latex]cos β[/latex] + [latex]sin α[/latex] [latex]sin β[/latex]

First, we will prove the difference formula for cosines. Let’s consider two points on the unit circle. See Figure 3. Point [latex]P[/latex] is at an angle [latex]α[/latex] from the positive x-axis with coordinates ([latex]cos α[/latex], [latex]sin α[/latex]) and point [latex]Q[/latex] is at an angle of [latex]β[/latex] from the positive x-axis with coordinates ([latex]cos β[/latex], [latex]sin β[/latex]). Note the measure of angle [latex]POQ[/latex] is [latex]α[/latex] − [latex]β[/latex].

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

We can find the distance from [latex]P[/latex] to [latex]Q[/latex] using the distance formula.

Then we apply the Pythagorean Identity and simplify.

Similarly, using the distance formula we can find the distance from [latex]A[/latex] to [latex]B[/latex].

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.

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.

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 quotient of the sum formulas for sine and cosine and simplifying. Recall, [latex]tan x[/latex] = [latex]\frac{sin x}{cos x}[/latex], [latex]cos x[/latex] ≠ 0.

Let’s derive the sum formula for tangent.

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

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 [latex]\frac{π}{2}[/latex], those two angles are complements, and the sum of the two acute angles in a right triangle is [latex]\frac{π}{2}[/latex], so they are also complements. In Figure 6, notice that if one of the acute angles is labeled as [latex]θ[/latex], then the other acute angle must be labeled ([latex]\frac{π}{2}[/latex] − [latex]θ[/latex]).

Notice also that [latex]sin θ[/latex] = [latex]cos[/latex] ([latex]\frac{π}{2}[/latex] − [latex]θ[/latex]): opposite over hypotenuse. Thus, when two angles are complementary, we can say that the sine of [latex]θ[/latex] equals the cofunction of the complement of [latex]θ[/latex]. 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 justified algebraically using the sum and difference formulas. For example, using:

we can write:

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 Solving Trigonometric Equations with Identities may help simplify the process of verifying an identity.

Media

Access these online resources for additional instruction and practice with sum and difference identities.

• Sum and Difference Identities for Cosine
• Sum and Difference Identities for Sine
• Sum and Difference Identities for Tangent

Section Exercises

Verbal

1. Explain the basis for the cofunction identities and when they apply.

2. Is there only one way to evaluate [latex]cos[/latex] ([latex]\frac{5π}{4}[/latex])? Explain how to set up the solution in two different ways, and then compute to make sure they give the same answer.

3. Explain to someone who has forgotten the even-odd properties of sinusoidal functions how the addition and subtraction formulas can determine this characteristic for [latex]f(x)[/latex] = [latex]sin (x)[/latex] and [latex]g(x)[/latex] = [latex]cos (x)[/latex]. (Hint: 0 − [latex]x[/latex] = −[latex]x[/latex])

Algebraic

For the following exercises, find the exact value.

4. [latex]cos[/latex] ([latex]\frac{7π}{12}[/latex])

5. [latex]cos[/latex] ([latex]\frac{π}{12}[/latex])

6. [latex]sin[/latex] ([latex]\frac{5π}{12}[/latex])

7. [latex]sin[/latex] ([latex]\frac{11π}{12}[/latex])

8. [latex]tan[/latex] (−[latex]\frac{π}{12}[/latex])

9. [latex]tan[/latex] ([latex]\frac{19π}{12}[/latex])

For the following exercises, rewrite in terms of [latex]sin[/latex] [latex]x[/latex] and [latex]cos[/latex] [latex]x[/latex].

10. [latex]sin[/latex] ([latex]x[/latex] + [latex]\frac{11π}{6}[/latex])

11. [latex]sin[/latex] ([latex]x[/latex] − [latex]\frac{3π}{4}[/latex])

12. [latex]cos[/latex] ([latex]x[/latex] − [latex]\frac{5π}{6}[/latex])

13. [latex]cos[/latex] ([latex]x[/latex] + [latex]\frac{2π}{3}[/latex])

For the following exercises, simplify the given expression.

14. [latex]csc[/latex] ([latex]\frac{π}{2}[/latex] − [latex]t[/latex])

15. [latex]sec[/latex] ([latex]\frac{π}{2}[/latex] − [latex]θ[/latex])

16. [latex]cot[/latex] ([latex]\frac{π}{2}[/latex] − [latex]x[/latex])

17. [latex]tan[/latex] ([latex]\frac{π}{2}[/latex] − [latex]x[/latex])

18. [latex]sin[/latex] ([latex]2x[/latex]) [latex]cos[/latex] ([latex]5x[/latex]) − [latex]sin[/latex] ([latex]5x[/latex]) [latex]cos[/latex] ([latex]2x[/latex])

19. [latex]\frac{tan (\frac{3}{2}x) − tan (\frac{7}{5}x)}{1 + tan (\frac{3}{2}x) tan (\frac{7}{5}x)}[/latex]

For the following exercises, find the requested information.

20. Given that [latex]sin[/latex] [latex]a[/latex] = [latex]\frac{2}{3}[/latex] and [latex]cos[/latex] [latex]b[/latex] = −[latex]\frac{1}{4}[/latex], with [latex]a[/latex] and [latex]b[/latex] both in the interval [[latex]\frac{π}{2}[/latex] , [latex]π[/latex]), find [latex]sin[/latex] ([latex]a[/latex] + [latex]b[/latex]) and [latex]cos[/latex] ([latex]a[/latex] − [latex]b[/latex]).

21. Given that [latex]sin[/latex] [latex]a[/latex] = [latex]\frac{4}{5}[/latex] and [latex]cos[/latex] [latex]b[/latex] = −[latex]\frac{1}{3}[/latex], with [latex]a[/latex] and [latex]b[/latex] both in the interval [[latex]0[/latex], [latex]π[/latex]), find [latex]sin[/latex] ([latex]a[/latex] − [latex]b[/latex]) and [latex]cos[/latex] ([latex]a[/latex] + [latex]b[/latex]).

For the following exercises, find the exact value of each expression.

22. [latex]sin[/latex] ([latex]cos^{−1}[/latex]([latex]0[/latex]) − [latex]cos^{−1}[/latex]([latex]\frac{1}{2}[/latex]))

23. [latex]cos[/latex] ([latex]cos^{−1}[/latex]([latex]\frac{\sqrt{2}}{2}[/latex]) + [latex]sin^{−1}[/latex]([latex]\frac{\sqrt{3}}{2}[/latex]))

24. [latex]tan[/latex] ([latex]sin^{−1}[/latex]([latex]\frac{1}{2}[/latex]) − [latex]cos^{−1}[/latex]([latex]\frac{1}{2}[/latex]))

Graphical

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical.

25. [latex]cos[/latex] ([latex]\frac{π}{2}[/latex] − [latex]x[/latex])

26. [latex]sin[/latex] ([latex]π[/latex] − [latex]x[/latex])

27. [latex]tan[/latex] ([latex]\frac{π}{3}[/latex] + [latex]x[/latex])

28. [latex]sin[/latex] ([latex]\frac{π}{3}[/latex] + [latex]x[/latex])

29. [latex]tan[/latex] ([latex]\frac{π}{4}[/latex] − [latex]x[/latex])

30. [latex]cos[/latex] ([latex]\frac{7π}{6}[/latex] + [latex]x[/latex])

31. [latex]sin[/latex] ([latex]\frac{π}{4}[/latex] + [latex]x[/latex])

32. [latex]cos[/latex] ([latex]\frac{5π}{4}[/latex] + [latex]x[/latex])

For the following exercises, use a graph to determine whether the functions are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think [latex]2x[/latex] = [latex]x[/latex] + [latex]x[/latex])

33. [latex]f(x)[/latex] = [latex]sin[/latex] ([latex]4x[/latex]) − [latex]sin[/latex] ([latex]3x[/latex]) [latex]cos[/latex] [latex]x[/latex], [latex]g(x)[/latex] = [latex]sin[/latex] [latex]x[/latex] [latex]cos[/latex] ([latex]3x[/latex])

34. [latex]f(x)[/latex] = [latex]cos[/latex] ([latex]4x[/latex]) + [latex]sin[/latex] [latex]x[/latex] [latex]sin[/latex] ([latex]3x[/latex]), [latex]g(x)[/latex] = −[latex]cos[/latex] [latex]x[/latex] [latex]cos[/latex] ([latex]3x[/latex])

35. [latex]f(x)[/latex] = [latex]sin[/latex] ([latex]3x[/latex]) [latex]cos[/latex] ([latex]6x[/latex]), [latex]g(x)[/latex] = −[latex]sin[/latex] ([latex]3x[/latex]) [latex]cos[/latex] ([latex]6x[/latex])

36. [latex]f(x)[/latex] = [latex]sin[/latex] ([latex]4x[/latex]), [latex]g(x)[/latex] = [latex]sin[/latex] ([latex]5x[/latex]) [latex]cos[/latex] [latex]x[/latex] − [latex]cos[/latex] ([latex]5x[/latex]) [latex]sin[/latex] [latex]x[/latex]

37. [latex]f(x)[/latex] = [latex]sin[/latex] ([latex]2x[/latex]), [latex]g(x)[/latex] = 2 [latex]sin[/latex] [latex]x[/latex] [latex]cos[/latex] [latex]x[/latex]

38. [latex]f(θ)[/latex] = [latex]cos[/latex] ([latex]2θ[/latex]), [latex]g(θ)[/latex] = [latex]cos[/latex] ([latex]2θ[/latex]) − [latex]sin[/latex] ([latex]2θ[/latex])

39. [latex]f(θ)[/latex] = [latex]tan[/latex] ([latex]2θ[/latex]), [latex]g(θ)[/latex] = [latex]\frac{tan θ}{1 + tan (2θ)}[/latex]

40. [latex]f(x)[/latex] = [latex]sin[/latex] ([latex]3x[/latex]) [latex]sin[/latex] [latex]x[/latex], [latex]g(x)[/latex] = [latex]sin^2[/latex] ([latex]2x[/latex]) [latex]cos^2 x[/latex] − [latex]cos^2[/latex] ([latex]2x[/latex]) [latex]sin[/latex] ([latex]2x[/latex])

41. [latex]f(x)[/latex] = [latex]tan[/latex] (−[latex]x[/latex]), [latex]g(x)[/latex] = [latex]\frac{tan x − tan(2x)}{1 − tan x tan (2x)}[/latex]

Extensions

For the following exercises, prove or disprove the statements.

42. [latex]tan[/latex] ([latex]u[/latex] + [latex]v[/latex]) = [latex]\frac{tan u + tan v}{1 − tan u tan v}[/latex]

43. [latex]tan[/latex] ([latex]u[/latex] − [latex]v[/latex]) = [latex]\frac{tan u − tan v}{1 + tan u tan v}[/latex]

44. [latex]\frac{tan (x + y)}{1 + tan x tan x}[/latex] = [latex]\frac{tan x + tan y}{1 − tan^2 x tan^2 y}[/latex]

45. If [latex]α[/latex], [latex]β[/latex], and [latex]γ[/latex] are angles in the same triangle, then prove or disprove [latex]sin[/latex] ([latex]α[/latex] + [latex]β[/latex]) = [latex]sin[/latex] [latex]γ[/latex].

46. If [latex]α[/latex], [latex]β[/latex], and [latex]γ[/latex] are angles in the same triangle, then prove or disprove [latex]tan[/latex] [latex]α[/latex] + [latex]tan[/latex] [latex]β[/latex] + [latex]tan[/latex] [latex]γ[/latex] = [latex]tan[/latex] [latex]α[/latex] [latex]tan[/latex] [latex]β[/latex] [latex]tan[/latex] [latex]γ[/latex].

For the following exercises, prove the identities provided.

47. [latex]tan[/latex] ([latex]x[/latex] + [latex]\frac{π}{4}[/latex]) = [latex]\frac{tan x + 1}{1 − tan x}[/latex]

48. [latex]\frac{tan (a + b)}{tan (a − b)}[/latex] = [latex]\frac{sin a cos a + sin b cos b}{sin a cos a − sin b cos b}[/latex]

59. [latex]\frac{cos(a + b)}{cos a cos b}[/latex]= 1 − [latex]tan[/latex] [latex]a[/latex] [latex]tan[/latex] [latex]b[/latex]

50. [latex]cos[/latex] ([latex]x[/latex] + [latex]y[/latex]) [latex]cos[/latex] ([latex]x[/latex] − [latex]y[/latex]) = [latex]cos^2[/latex] [latex]x[/latex] − [latex]sin^2[/latex] [latex]y[/latex]

51. [latex]\frac{cos (x + h) − cos x}{h}[/latex] = [latex]cos[/latex] [latex]x[/latex] [latex]\frac{cos h − 1}{h}[/latex] − [latex]sin[/latex] [latex]x[/latex] [latex]\frac{sin h}{h}[/latex]