6.5: Sum-to-Product and Product-to-Sum Identities

Learning Objectives

• Rewrite a product of sines and cosines as a sum or a difference.
• Express a sum or difference of sines and cosines as a product.
• Prove an identity using the Sum-to-Product and Product-to-Sum Identities.

You should commit to memory the identities we have covered up to this point. Specifically, you need to be able to recall the following identities without access to a note card.

Reciprocal Identities
Basic Identity	Alternate Form
\( \sin\left( \theta \right) = \dfrac{1}{\csc\left( \theta \right)} \)	\( \csc\left( \theta \right) = \dfrac{1}{\sin\left( \theta \right)} \)
\( \cos\left( \theta \right) = \dfrac{1}{\sec\left( \theta \right)} \)	\( \sec\left( \theta \right) = \dfrac{1}{\cos\left( \theta \right)} \)
\( \tan\left( \theta \right) = \dfrac{1}{\cot\left( \theta \right)} \)	\( \cot\left( \theta \right) = \dfrac{1}{\tan\left( \theta \right)} \)

Ratio Identities
Basic Identity	Alternate Form
\( \tan\left( \theta \right) = \dfrac{\sin\left( \theta \right)}{\cos\left( \theta \right)}\)	\( \cot\left( \theta \right) = \dfrac{\cos\left( \theta \right)}{\sin\left( \theta \right)}\)

Pythagorean Identities
Basic Identity	Alternate Forms
\( \cos^2\left( \theta \right) + \sin^2\left( \theta \right) = 1 \)	\( \cos^2\left( \theta \right) = 1 - \sin^2\left( \theta \right) \) and \( \cos\left( \theta \right) = \pm \sqrt{1 - \sin^2\left( \theta \right)} \)
	\( \sin^2\left( \theta \right) = 1 - \cos^2\left( \theta \right) \) and \( \sin\left( \theta \right) = \pm \sqrt{1 - \cos^2\left( \theta \right)} \)
\( 1 + \tan^2\left( \theta \right) = \sec^2\left( \theta \right) \)	\( \tan^2\left( \theta \right) = \sec^2\left( \theta \right) - 1 \)
\( \cot^2\left( \theta \right) + 1 = \csc^2\left( \theta \right) \)	\( \cot^2\left( \theta \right) = \csc^2\left( \theta \right) - 1 \)

Symmetry Identities
Basic Identity	Alternate Form
\( \sin\left( -\theta \right) = -\sin\left( \theta \right)\)	\( \csc\left( -\theta \right) = -\csc\left( \theta \right)\)
\( \cos\left( -\theta \right) = \cos\left( \theta \right)\)	\( \sec\left( -\theta \right) = \sec\left( \theta \right)\)
\( \tan\left( -\theta \right) = -\tan\left( \theta \right)\)	\( \cot\left( -\theta \right) = -\cot\left( \theta \right)\)

Cofunction Identities
Basic Identity	Alternate Form
\( \sin\left( \theta \right) = \cos\left( \dfrac{\pi}{2} - \theta \right) \)	\( \cos\left( \theta \right) = \sin\left( \dfrac{\pi}{2} - \theta \right) \)
\( \sec\left( \theta \right) = \csc\left( \dfrac{\pi}{2} - \theta \right) \)	\( \csc\left( \theta \right) = \sec\left( \dfrac{\pi}{2} - \theta \right) \)
\( \tan\left( \theta \right) = \cot\left( \dfrac{\pi}{2} - \theta \right) \)	\( \cot\left( \theta \right) = \tan\left( \dfrac{\pi}{2} - \theta \right) \)

Sum and Difference of Angles Identities
\( \sin\left( \alpha \pm \beta \right) = \sin\left( \alpha \right)\cos\left( \beta \right) \pm \cos\left( \alpha \right)\sin\left( \beta \right) \)
\( \cos\left( \alpha \pm \beta \right) = \cos\left( \alpha\right)\cos\left( \beta \right) \mp \sin\left( \alpha \right) \sin\left( \beta \right) \)

Double-Angle Identities
\( \sin\left( 2\theta \right) = 2 \sin\left( \theta \right)\cos\left( \theta \right) \)
\[ \begin{array}{rcl} \cos\left( 2 \theta \right) & = & \cos^2\left( \theta \right) - \sin^2\left( \theta \right) \\ & = & 2\cos^2\left( \theta \right) - 1 \\ & = & 1 - 2 \sin^2\left( \theta \right) \\ \end{array} \nonumber \]

Power Reduction Identities
\( \sin^2\left( \theta \right) = \dfrac{1 - \cos\left( 2\theta \right)}{2} \)
\( \cos^2\left( \theta \right) = \dfrac{1 + \cos\left( 2\theta \right)}{2} \)

Half-Angle Identities
\( \sin\left( \dfrac{\theta}{2} \right) = \pm \sqrt{\dfrac{1 - \cos\left( \theta \right)}{2}} \)
\( \cos\left( \dfrac{\theta}{2} \right) = \pm \sqrt{\dfrac{1 + \cos\left( \theta \right)}{2}} \)

You should note that the tangent identities are not listed for our recent formulas. This is because they are rarely used and, if needed, can be derived from the identities for sine and cosine.

The Sum-to-Product and Product-to-Sum Identities

Our next batch of identities, the Product-to-Sum Identities,¹ can be easily verified by expanding each of the right-hand sides per the Sum and Difference Identities (we leave the details as exercises). They are of particular use in Calculus, and we list them here for reference.

Theorem: Product-to-Sum Identities

For all angles \(\alpha\) and \(\beta\),\[ \begin{array}{rcl}
\sin(\alpha)\sin(\beta) & = & \dfrac{1}{2} \left[ \cos(\alpha - \beta) - \cos(\alpha + \beta)\right] \\
\\
\cos(\alpha)\cos(\beta) & = & \dfrac{1}{2} \left[ \cos(\alpha - \beta) + \cos(\alpha + \beta)\right] \\
\\
\sin(\alpha)\cos(\beta) & = & \dfrac{1}{2} \left[ \sin(\alpha - \beta) + \sin(\alpha + \beta)\right] \\
\end{array} \nonumber \]

The final set of identities in this chapter, the Sum-to-Product Identities, are related to the Product-to-Sum Identities. These are easily verified using the Product-to-Sum Identities, and as such, their proofs are left as exercises.

Theorem: Sum-to-Product Identities

For all angles \(\alpha\) and \(\beta\),\[ \begin{array}{rcl}
\sin(\alpha) \pm \sin(\beta) & = & 2 \sin\left( \dfrac{\alpha \pm \beta}{2}\right)\cos\left( \dfrac{\alpha \mp \beta}{2}\right) \\
\\
\cos(\alpha) + \cos(\beta) & = & 2 \cos\left( \dfrac{\alpha + \beta}{2}\right)\cos\left( \dfrac{\alpha - \beta}{2}\right) \\
\\
\cos(\alpha) - \cos(\beta) & = & - 2 \sin\left( \dfrac{\alpha + \beta}{2}\right)\sin\left( \dfrac{\alpha - \beta}{2}\right) \\
\end{array} \nonumber \]

Do Not Memorize These Identities!

I cannot think of any situation where committing these identities to memory is beneficial. It's enough to know they exist and when they are needed.

Example \( \PageIndex{1} \)

1. Write \(\cos(2\theta)\cos(6\theta)\) as a sum.
2. Write \(\sin(4\theta)\cos(2\theta)\) as a sum.
3. Write \(\sin(\theta) - \sin(3\theta)\) as a product.

Solutions

1. Identifying \(\alpha = 2\theta\) and \(\beta = 6\theta\), we find\[\begin{array}{rcl}
   \cos(2\theta)\cos(6\theta) & = & \dfrac{1}{2} \left[ \cos(2\theta - 6\theta) + \cos(2\theta + 6\theta)\right] \\
   \\
   & = & \dfrac{1}{2} \cos(-4\theta) + \dfrac{1}{2}\cos(8\theta) \\
   \\
   & = & \dfrac{1}{2} \cos(4\theta) + \dfrac{1}{2} \cos(8\theta), \\
   \end{array}\nonumber\]where the last equality is courtesy of the even identity for cosine, \(\cos(-4\theta) = \cos(4\theta)\).

2. Write the formula for the product of sine and cosine. Then, substitute the given values into the formula and simplify.\[\begin{array}{rrcl}
   & \sin\left(\alpha\right)\cos\left(\beta\right) & = & \dfrac{1}{2}[ \sin(\alpha+\beta)+\sin(\alpha-\beta) ] \\
   \\
   \implies & \sin(4\theta)\cos(2\theta) & = & \dfrac{1}{2}[\sin(4\theta+2\theta)+\sin(4\theta-2\theta)] \\
   \\
   & & = & \dfrac{1}{2}[\sin(6\theta)+\sin(2\theta)] \\
   \end{array} \nonumber \]

3. Identifying \(\alpha = \theta\) and \(\beta = 3\theta\) yields\[\begin{array}{rcl}
   \sin(\theta) - \sin(3\theta) & = & 2 \sin\left( \dfrac{\theta - 3\theta}{2}\right)\cos\left( \dfrac{\theta + 3\theta}{2}\right) \\
   \\
   & = & 2 \sin\left( -\theta \right)\cos\left( 2\theta \right) \\
   \\
   & = & -2 \sin\left( \theta \right)\cos\left( 2\theta \right), \\
   \end{array}\nonumber\]where the last equality is courtesy of the odd identity for sine, \(\sin(-\theta) = -\sin(\theta)\).

The reader is reminded that all of the identities presented in this section, which regard the circular functions as functions of angles (in radian measure), apply equally well to the circular (trigonometric) functions considered as functions of real numbers.

Checkpoint \(\PageIndex{1}\)

1. Write the product as a sum or difference.\[\sin(x + y)\cos(x - y)\nonumber \]
2. Write the sum as a product.\[\sin(3\theta)+\sin(\theta)\nonumber \]

Answer

1. \(\dfrac{1}{2}\left(\sin\left( 2x \right) + \sin\left( 2y \right)\right)\)
2. \(2\sin(2\theta)\cos(\theta)\)

Using the Sum-to-Product and Product-to-Sum Identities to Evaluate Trigonometric Functions

There are times when obtaining exact values of trigonometric functions is easier when using the Sum-to-Product and Product-to-Sum Identities. Again, you are not expected to memorize these identities, but you are expected to be able to recognize when they might be of use.

Example \(\PageIndex{2}\)

Find the exact value of \(\cos(15^{ \circ })−\cos(75^{ \circ })\).

Solution

We could tackle this problem by using a Half-Angle Identity for the first term and a Sum of Angles Identity for the second; however, recognizing that this is a difference of cosines also gives us the clue that we could use a Sum-to-Product Identity. \[\begin{array}{rclcl}
\cos\left(15^{\circ}\right)-\cos\left(75^{\circ}\right) & = & -2\sin\left(\dfrac{15^{\circ}+75^{\circ}}{2}\right) \sin\left(\dfrac{15^{\circ}-75^{\circ}}{2}\right) & \quad & \left( \text{Sum-to-Product Identity} \right)\\
\\
& = & -2\sin\left(45^{\circ}\right) \sin\left(-30^{\circ}\right) & & \\
\\
& = & -2 \left(\dfrac{1}{\sqrt{2}}\right)\left(-\dfrac{1}{2}\right) & & \\
\\
& = & \dfrac{1}{\sqrt{2}}
\end{array} \nonumber \]

Checkpoint \(\PageIndex{2}\)

Find the exact value of \(\cos\left(\dfrac{11\pi}{12}\right)\cos\left(\dfrac{\pi}{12}\right)\).

Answer

\(\dfrac{−2−\sqrt{3}}{4}\)

Revisiting Proofs of Identities

While proofs of identities that require Sum-to-Product or Product-to-Sum Identities are no more challenging than proofs involving the other trigonometric identities, using these newer identities less frequently makes recognizing the need for them more difficult. Try your best to remember that when all else fails, you might need to use a Sum-to-Product or Product-to-Sum Identity.

Example \(\PageIndex{3}\)

Prove the identity.\[\dfrac{\cos(4t)−\cos(2t)}{\sin(4t)+\sin(2t)}=−\tan (t)\nonumber \]

Solution

We will start with the left side, the more complicated side of the equation, and rewrite the expression until it matches the right side.\[\begin{array}{rclcl}
\text{LHS} & = & \dfrac{\cos(4t)-\cos(2t)}{\sin(4t)+\sin(2t)} & & \\
\\
& = & \dfrac{-2 \sin\left(\dfrac{4t+2t}{2}\right) \sin\left(\dfrac{4t-2t}{2}\right)}{2 \sin\left(\dfrac{4t+2t}{2}\right) \cos\left(\dfrac{4t-2t}{2}\right)} & \quad & \left( \text{Sum-to-Product Identities} \right) \\
\\
& = & \dfrac{-2 \sin(3t)\sin (t)}{2 \sin(3t)\cos (t)} & & \\
\\
& = & -\dfrac{\sin (t)}{\cos (t)} & & \\
\\
& = & -\tan(t) & & \\
\\
& = & \text{RHS} & & \\
\end{array} \nonumber \]

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Footnotes

¹ These are also known as the Prosthaphaeresis Formulas and have a rich history. I recommend that you conduct some research on them as your schedule allows.

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Homework

Basic Skills

For Problems 1 - 6, write the given product as a sum.

1. \(\cos(3\theta)\cos(5\theta)\)

2. \(\sin(2\theta)\sin(7\theta)\)

3. \(\sin(9\theta)\cos(\theta)\)

4. \(\cos(2\theta) \cos(6\theta)\)

5. \(\sin(3\theta) \sin(2\theta)\)

6. \(\cos(\theta) \sin(3\theta)\)

For Problems 7 - 12, write the given sum as a product.

7. \(\cos(3\theta) + \cos(5\theta)\)

8. \(\sin(2\theta) - \sin(7\theta)\)

9. \(\cos(5\theta) - \cos(6\theta)\)

10. \(\sin(9\theta) - \sin(-\theta)\)

11. \(\sin(\theta) + \cos(\theta)\)

12. \(\cos(\theta) - \sin(\theta)\)

For Problems 13 - 15, find the exact values of the trigonometric expressions.

13. \( 2 \cos\left( \dfrac{\pi}{5} \right) \cos\left( \dfrac{\pi}{3} \right)\)

14. \( \sin\left( \dfrac{\pi}{12} \right) - \sin\left( \dfrac{7\pi}{12} \right) \)

15. \( \cos\left( \dfrac{5\pi}{12} \right) + \cos\left( \dfrac{7\pi}{12} \right)\)

Proofs

16. Verify the Product to Sum Identities.

17. Verify the Sum to Product Identities.

For Problems 18 - 23, prove the identity.

18. \(\dfrac{\sin(x) + \sin(y)}{\cos(x) + \cos(y)} = \tan\left(\dfrac{x + y}{2}\right)\)

19. \(\dfrac{\sin(x) - \sin(y)}{\cos(x) - \cos(y)} = \cot\left(\dfrac{x + y}{2}\right)\)

20. \(\sin(x) + \sin(2x) + \sin(3x) = 4\cos\left(\dfrac{x}{2}\right) \cos(x) \sin\left(\dfrac{3x}{2}\right)\)

21. \(\cos(x) + \cos(2x) + \cos(3x) = 4\cos\left(\dfrac{x}{2}\right) \cos(x) \cos\left(\dfrac{3x}{2}\right) - 1\)

22. \(\dfrac{\cos(x) - \cos(3x)}{\sin(3x) - \sin(x)} = \tan(2x)\)

23. \(-\tan(4x) = \dfrac{\cos(3x) - \cos(5x)}{\sin(3x) - \sin(5x)} \)