Integration Using Trigonometric Identities

Key IdeaIntegration using trigonometric identities transforms complex or seemingly impossible trigonometric integrals into manageable forms by using algebraic relationships to rewrite the integrand before integrating. This method is especially useful when dealing with powers of trigonometric functions, products of different trig functions, or expressions that perfectly fit standard u-substitution patterns.

Contents

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domains. They allow rewriting expressions in different but equivalent forms. Understanding identities like double-angle and half-angle formulas is essential to apply power-reducing formulas correctly. Integrals of polynomials of the trigonometric functions sin x, cos x, tan x and so on, are generally evaluated by using a combination of simple substitutions and trigonometric identities.

Key Concepts

Pythagorean Identities

Raising trigonometric functions to powers, such as sin⁴x, involves repeated multiplication. To simplify or integrate such expressions, it is necessary to rewrite them using identities that reduce the power, often by expressing higher powers in terms of lower powers or multiple angles. The fundamental Pythagorean identities allow you to swap between squares of different functions to facilitate u‑substitution.

  • sin2(x) + cos2(x) = 1
  • tan2(x) + 1 = sec2(x)
  • cot2(x) + 1 = csc2(x)

Reduction Identities for Powers

  • Even Powers of Sine and Cosine: Use half-angle formulas like sin2(x) = (1 – cos(2x))/2 and cos2(x) = (1 + cos(2x))/2.
  • Higher Even Powers: Rewrite as squared terms, such as sin4(x) = (sin2(x))2, then expand using the half-angle identity.
  • Odd Powers of Sine and Cosine: Separate one factor to pair with a differential or use a Pythagorean identity like sin2(x) + cos2(x) = 1.
  • Powers of Tangent and Secant: Pair sec2(x) with the derivative of tan(x), or convert tangents to sines and cosines.

How to Use Them

If you are integrating a product like ∫ sinn(x) cosm(x) dx and one of the exponents is an odd integer:

  1. Factor off one power of the odd function.
  2. Convert the remaining even powers using the appropriate Pythagorean identity.
  3. Use u-substitution with the remaining factored function.

Example ∫ sin3(x) dx

  1. Factor ∫ sin2(x) sin(x) dx
  2. Convert ∫ (1 – cos2(x)) sin(x) dx
  3. Let u = cos(x) and du = -sin(x) dx

\int{-\left( 1-u^2 \right)}du\,\,=\,\,\frac{u^3}{3}-u\,\,+\,\,C\,\,=\,\,\frac{\cos ^3\left( x \right)}{3}\,\,-\,\,\cos \left( x \right) \,\,+\,\,C

Half-Angle (Power-Reducing Formulas)

When dealing with even powers of sines and cosines (such as sin2(x) or cos2(x)), standard integration rules do not apply directly. Power-reducing formulas allow you to rewrite these terms into expressions with a power of 1, which are easy to integrate.

Power Reducing Identities – SOM

How to Use Them

Substitute these identities directly for any sin2(x) or cos2(x) before evaluating the integral.

Example ∫ cos2(x) dx

  1. Convert ∫ (1 – cos2(x))/2 dx
  2. Integrate

\frac{1}{2}\int{\left( 1+\cos \left( 2x \right) \right) dx}q\,=\,\frac{1}{2}\left( x+\frac{1}{2}\sin \left( 2x \right) \right) \,+\,C\,=\frac{x}{2}+\frac{\sin \left( 2x \right)}{4}\,+\,C

Product-to-Sum Identities

If your integrand is a product of different trigonometric functions with varying frequencies (e.g., ∫ sin(ax)cos(bx) dx), you can use product-to-sum identities to rewrite them as a sum/difference, which can be integrated term by term.

\sin(A)\cos(B) = \frac{1}{2}[\sin(A+B) + \sin(A-B)]

\sin(A)\sin(B) = \frac{1}{2}[\cos(A-B) - \cos(A+B)]

\cos(A)\cos(B) = \frac{1}{2}[\cos(A-B) + \cos(A+B)]

Double Angle Identities

Double-angle identities are used to simplify quotients or unpack expressions that are difficult to evaluate.

  • sin(2x) = 2sin(x)cos(x)
  • cos(2x) = cos2(x) – sin2(x) = 2cos2(x) – 1 = 1 – 2sin2(x)

Example ∫ sin(2x)/cos(x) dx

1. Convert

\int \frac{2\sin(x)\cos(x)}{\cos(x)} \,dx

2. Simplify & Integrate

\int 2\sin(x) \,dx = -2\cos(x) + C

Practice Problems

Dawkins, Paul. “Calculus II.” Integrals Involving Trig Functions (Practice Problems). Paul’s Online Notes, November 16, 2022. https://tutorial.math.lamar.edu/Problems/CalcII/IntegralsWithTrig.aspx.

References

“In Exercises 35–38, Use the Power-Reducing Formulas to Rewrite – Blitzer 3rd Edition Ch 3 Problem 3.3.35.” Pearson+, Accessed July 17, 2026. https://www.pearson.com/channels/trigonometry/textbook-solutions/blitzer-trigonometry-3rd-edition-9780137316601/ch-03-trigonometric-identities-and-equations/in-exercises-3538-use-the-power-reducing-formulas-to-rewrite-each-expression-as-.

“Summary of Trigonometric Integrals.” Calculus II. Module 3: Techniques of Integration. lumen, Accessed July 17, 2026. https://courses.lumenlearning.com/calculus2/chapter/summary-of-trigonometric-integrals/.

“Trigonometric Integrals.” Fiveable, n.d. Accessed July 17, 2026. https://fiveable.me/calc-ii/unit-3/2-trigonometric-integrals/study-guide/2LbNHGIDZZfkXFnE.

Feldman, Joel, Andrew Rechnitzer and Elyse Yeager. Trigonometric Integrals. n.d. CLP-2 Integral Calculus. Accessed July 17, 2026. https://personal.math.ubc.ca/~CLP/CLP2/clp_2_ic/sec_trigint.html.

Additional Reading

Amidi, Shervine. “CME 102 – Trigonometry Refresher.” Stanford University. Accessed July 17, 2026. https://stanford.edu/~shervine/teaching/cme-102/trigonometry/.

Dawkins, Paul. “Calculus II.” Integrals Involving Trig Functions. Paul’s Online Notes, November 16, 2022. https://tutorial.math.lamar.edu/classes/calcii/integralswithtrig.aspx.

“How to Find Solutions in an Interval for an Equation with Sine & Cosine Using Double-Angle Identities.” n.d. study.com, Accessed July 17, 2026. https://study.com/skill/learn/how-to-find-solutions-in-an-interval-for-an-equation-with-sine-cosine-using-double-angle-identities-explanation.html.

“Integration of Trigonometric Functions.” GeeksforGeeks, April 25, 2021. https://www.geeksforgeeks.org/maths/integration-of-trigonometric-functions/.

“Integration using trig identities or a trig substitution.” mathcentre, 2009. https://www.mathcentre.ac.uk/resources/uploaded/mc-ty-intusingtrig-2009-1.pdf.

Simpson, Roy. “9.4: Half-Angle and Power Reduction Identities.” LibreTexts, July 8, 2025. https://math.libretexts.org/Courses/Cosumnes_River_College/Math_375%3A_Pre-Calculus/09%3A_Analytic_Trigonometry/9.04%3A_Half-Angle_and_Power_Reduction_Identities.

Spong, Mark. “Double, Half, and Power Reducing Identities.” CK-12 Foundation, November 27, 2023. https://flexbooks.ck12.org/cbook/ck-12-precalculus-concepts-2.0/section/6.4/primary/lesson/double-half-and-power-reducing-identities-pcalc/.

Stewart, James. “Trigonometric Integrals.” Stewart Calculus, Accessed July 17, 2026. https://www.stewartcalculus.com/data/CALCULUS%20Concepts%20and%20Contexts/upfiles/3c3-TrigonometIntegrals_Stu.pdf.

“Trig. Integrals.” xaktly, Accessed July 17, 2026. https://xaktly.com/IntegrationByTrigSubst.html.

[ ℰ ] “Trigonograph.” Mathematical Mysteries, January 1, 2024. https://mathematicalmysteries.org/trigonograph/.

“Trigonograph” is this author’s name for a geometric diagram that illustrates a trigonometric relation or concept, ideally in a way that makes things “obvious”. The term is a back-construction of the word-playful trigonography, which could be taken to mean “the art of trigonometric visualization”.

“Trigonometric Integrals.” xaktly, Accessed July 17, 2026. https://xaktly.com/TrigonometricIntegrals.html.

“Trigonometric Fundamental Identities: Complete Guide.” StudyMathNow, October 7, 2025. https://studymathnow.com/trigonometry/trigonometry-formulas/fundamental-identities-in-trigonometry/.

“Trigonometry Formulas List with Examples [PDF].” StudyMathNow, August 9, 2025. https://studymathnow.com/trigonometry/trigonometry-formulas/.

Whitney, Earl. “Math Handbook of Formulas, Processes and Tricks.” Version 2.4. Mathguy, December 17, 2023. https://www.mathguy.us/Handbooks/TrigonometryHandbook.pdf

Videos


Integrating with Trig Identities

 

For tips on handling odd powers and more advanced trig identities.

How to use well known (formula sheet) trigonometric identities to integrate some trigonometric expressions that are otherwise unfamiliar to us.

 

Trigonometric Integrals in Calculus

 

For a visual breakdown of how to integrate an expression using trigonometric identities.

Welcome to our enlightening YouTube video that dives deep into the world of trigonometric integrals in calculus, uncovering the art of integrating functions involving trigonometric expressions. In this illuminating tutorial, we take you on a journey through the intricacies of trigonometric integrals, equipping you with the techniques to conquer these complex integrals with confidence.

Whether you’re a calculus enthusiast or a student aiming to master integration techniques, this video is your ultimate guide. Our expert instructor demystifies the complexities, ensuring you grasp the essence of trigonometric integrals and their role in solving diverse mathematical problems.

Join us as we explore the universe of trigonometric integrals, from basic forms to more intricate expressions involving trigonometric identities and substitutions. With clear explanations and illustrative examples, we guide you through the steps to tackle various types of trigonometric integrals, empowering you to handle even the most challenging cases.

Through interactive visuals and real-world scenarios, you’ll develop the skills to confidently integrate trigonometric functions, gaining a deep understanding of their behavior and their application in physics, engineering, and other fields. You’ll learn how to leverage trigonometric identities and clever substitutions to simplify integrals and unravel their solutions.

Don’t let the complexities of trigonometric integrals intimidate you any longer. Join us for this enlightening tutorial and unlock the power to master trigonometric integrals, enhancing your integration toolkit and elevating your calculus skills. Hit that play button now and embark on your journey to becoming a master of integration and calculus!

 


Evaluating Trigonometric Integrals | Powers of Sine, Cosine, Tangent & More! | Math with Professor V

 

Updated video lecture on how to evaluate trigonometric integrals. Clear outlining of the various cases, how to use trigonometric identities and u-substitution. Powers of sine, cosine, tangent, and secant explained.

 

Calculus 2: Trigonometric Integrals (Video #2) | Math with Professor V

 

Overview and lots of examples of how to evaluate trigonometric integrals. Clear outlining of the various cases, how to use trigonometric identities and u-substitution. Powers of sine, cosine, tangent, and secant explained.

 


Integral of the Day 4.17.25 | Integration Challenge: Trig Identities Galore! Math with Professor V

 


Trigonometry: Introduction to Identities (Section 1.4) | Math with Professor V

An introduction to identities involving the trigonometric functions, including the reciprocal identities and Pythagorean Identities. Using the identities to find specific trigonometric values of given angles, and understanding how to determine the sign of a trigonometric function based on info provided.

[ ℰ ] This exceptional reference is highly recommended for your consideration.

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