Authentic Lessons for 21st Century Learning

Trig Identities, Part 1

Reciprocal, Quotient, and Pythagorean Identities

Michell Eike, Sherry Franklin | Published: August 17th, 2023 by K20 Center

  • Grade Level Grade Level 11th, 12th
  • Subject Subject Mathematics
  • Course Course Precalculus
  • Time Frame Time Frame 90-120 minutes
  • Duration More 2-3 class periods


Students will prove the reciprocal, quotient, and Pythagorean trigonometric identities. They will then use those identities to practice simplifying and verifying trigonometric identities during this spy-themed lesson. This lesson is an introduction to identities and is the first lesson of four in a “Trig Identities” lesson series.

Essential Question(s)

How do we use trigonometric identities?



Students recall special angles and find patterns between trigonometric expressions.


Students discover the three Pythagorean trigonometric identities.


Students complete guided notes with the class and learn how to simplify and verify trigonometric identities using reciprocal, quotient, and Pythagorean identities.


Students practice simplifying and verifying trigonometric identities through a card game.


Students reflect and summarize their learning while sharing advice for peers.


  • Lesson Slides (attached)

  • Gathering Intel handout (attached; one per pair; printed front only)

  • Secret Agent Pythagoras handout (attached; one per student; printed front only)

  • Intel Notes handout (attached; one per student; printed front/back)

  • Intel Notes (Model Notes) document (attached; for teacher use)

  • Trig Spies Cards (attached; one set per group; printed front/back)

  • Trig Spies Record Sheet handout (attached; one per group; printed front only)

  • Trig Spies Record Sheet (Sample Responses) document (attached; for teacher use)

  • Pencil

  • Paper

  • Green plastic cups (1 per pair)

  • Yellow plastic cups (1 per pair)

  • Red plastic cups (1 per pair)


20 Minute(s)

Introduce the lesson using the attached Lesson Slides. Slide 3 displays the lesson’s essential question. Slide 4 identifies the lesson’s learning objectives. Review each of these with the class to the extent you feel necessary.

Ask students to find a partner or assign pairs. Display slide 5 and pass out the Gathering Intel handout. Introduce the students to the I Notice, I Wonder instructional strategy and help set the scene for their “spy mission” lesson by telling them that they are a “covert operative” assigned with the task of gathering intel.

Assign each pair one of the following angle measures to investigate: 30°, 45°, 60°, π/6, π/4, or π/3. This can be done a number of ways: the easiest is to start in one corner of the room and assign the first pair 30°, the next pair 45°, …, this pair π/3, then the next pair 30°, and continue to repeat the 6 angle measures. The importance is that students are flexible with degrees and radians.

Instruct students to write their assigned angle measure at the top of their handout. Explain to them that they are to use that angle measure to complete the table.

Once they have completed the table, have pairs review their table and look for patterns. Have students write what they notice and what they wonder on their handout.

Display slide 6 and pose the following question to the students: Sine and Cosecant seem to be conspiring. What is their relationship?

Ask for volunteers to share anything they noticed. Use the hidden slide 7 to help move students from noticing the relationship between sine and cosecant for their angle to proving that the sine and cosecant are reciprocals.

Show slide 8 and pose the following question to the students: Cosine and Secant are plotting together. What is their connection?

Ask for volunteers to share anything they noticed. Use the hidden slide 9 to help move students from noticing the relationship between cosine and secant for their angle to proving that the cosine and secant are reciprocals.

Move to slide 10 and pose the following question to the students: Tangent and Cotangent seem to be scheming. What is their alliance?

Ask for volunteers to share anything they noticed. Use the hidden slide 11 to help move students from noticing the relationship between tangent and cotangent for their angle to proving that the tangent and cotangent are reciprocals.

Display slide 12 and pose the following question to the students: Sine and Cosecant seem to be in cahoots with Tangent and Cotangent but how?

Ask for volunteers to share anything they noticed. Use the hidden slides 13-14 to help move students from noticing the relationships between sine and cosine with tangent and sine and cosine with cotangent for their angle to proving those relationships.


25 Minute(s)

Transition to slide 15 and give each student a copy of the attached Secret Agent Pythagoras handout. Explain to the class that they are now “intelligence analysts” and need to look closely at the Pythagorean Theorem and see what they can uncover.

Display slide 16 and introduce the Try It, Talk It, Color It, Check It instructional strategy and provide each pair with a stack of three cups: one green, one yellow, and one red. Preview the activity by explaining that they are to begin by trying step 1 on their own, then compare their work with their partner, then use the stack of cups to indicate their confidence level of their final answer, then check their work. They will repeat this process for each of the four steps and work through this handout step-by-step as a class.

Let students know that the green cup means "We got it and can teach others!" The yellow cup indicates "We are a bit uncertain." And the red cup means "We need help. We may be wrong." Have each group start with the yellow cup on top, so that it is easy to see when there is a change.

Display slide 17 and instruct students to complete step 1 on their own. Then have them compare their answers, discuss their work, and agree upon a final result. Direct pairs to indicate their confidence level with their stack of cups: if they feel confident in their work to put the green cup on top, if they are unsure to put the yellow on top, and if they are completely lost to put the red cup on top.

Transition to slide 18 and have students check their work for step 1. Give students time to correct their work and ask questions as needed. As students are checking their work, assist pairs who have a red or yellow cup on top to ensure everyone is understanding before moving to the next step.

Repeat this process for step 2. Use slides 19-20. For step 3, use slides 21-22. For step 4, use slides 23-24. On completion of the handout, display slide 25 and share with the class the truth: We derived three trigonometric equations from the Pythagorean Theorem.


20 Minute(s)

Transition to slide 26. Give each student a copy of the Intel Notes handout.

Explain to the class that the equations they have seen during this lesson are known as “trigonometric identities.” Give students the definition of “identity,” which is on the slide.

Show slide 27 and ask the class to identify which of the given equations is an identity.

Direct students’ attention to their handout and explain to students that they already proved the reciprocal, quotient, and Pythagorean identities earlier in the lesson.

Display slide 28 and explain how to read and write the shorthand notation for trigonometric expressions raised to the second power. Help students see this shorthand notation in the Pythagorean identities on their handout.

Complete example 1 as a class. Use the hint of rewriting expressions using sine and cosine to simplify the given expression.

Challenge students to try example 2 on their own. While students work, monitor progress by circulating the room. Depending on time, write the steps on the board slowly so that students can check their work as they go or have a volunteer go to the board to share their work.

Direct students’ attention to the back of their handout and explain how similar, yet different, verifying trigonometric identities is from simplifying trigonometric identities. Let students know that they can start with either side of the equation, but it is recommended—as it is often simpler—to make the complicated side look like the less complicated side.

Based on this advice, direct students to start example 3 on their own. After a few minutes, facilitate a brief discussion having students share why they started on the side they started and what their first couple of steps look like. Use student responses to determine whether the class is ready to finish example 3 on their own or whether they need to work through it together as a class.

Once finished, have students add the handout to their math notebook if that is a classroom norm.


20 Minute(s)

Display slide 29 and direct students to get into small groups of 2-3 or assign groups. Students could continue to work in the same pairs as earlier in the lesson or use this time to have students work with someone they have not yet worked with during this lesson. Working with different peers fosters the development of academic vocabulary and encourages students to consider different approaches to a problem.

Let students know that they are now “spymasters” and responsible for selecting agents for the next mission. Give each group a copy of the attached Trig Spies Record Sheet and a set of Trig Spies Cards (containing 6 Undercover Agents cards and 6 Deep Cover Agents cards). Direct students to place each stack of cards face down so that the names of the cards with their icons are facing up. Make sure each student has a piece of scratch paper and writing tool for this game. Remind students to not draw on the cards, especially if you plan to reuse the cards.

Display slide 30 and ask each team to check that they have everything they need for the mission (card game):

  • 2 Stacks of Cards

  • Scratch Paper

  • Writing Utensil

  • Trig Spies Record Sheet

Display slide 31 and go over the directions for the card game. Explain that teams will pick a stack and flip over the top card. On their scratch paper, each student works independently to complete the question, and then they compare their work with their teammate(s). Once everyone on the team agrees, they write the card letter (A, B, C, etc.), the identity they used, and the points they earned on the team’s Trig Spies Record Sheet.

Show slide 32 and explain that each stack is worth different points. The cards in the Undercover Agent stack are worth 3-4 points; the Deep Cover Agent cards are worth 5-6 points and are more challenging. Share that they will have 15 minutes to earn as many points as they can. So, they need to select their agents wisely.

Transition to slide 33, begin the 15-minute timer, and tell the teams to begin.

Once the timer expires, ask students to add up their points. Then instruct students to put their name on their scratch paper and staple everyone’s scratch paper to their group’s Trig Spies Record Sheet. Collect these papers to assess students’ learning. Use the Trig Spies Record Sheet (Sample Responses) document for possible student responses.

Quickly skim the papers and announce the highest score, but not which team achieved that score. This will give students the opportunity to self-reflect and compare their score with the highest. If the class really wants to know which team earned the most points, remind them that you do not want to reveal any secret identities in the middle of such an important mission.


5 Minute(s)

Move to slide 34. Inform students that their expertise is needed for a new mission, so they need to pass along their knowledge to the new spymaster taking over. Have students write 2-3 pieces of advice for the new spymaster. Students can write their advice on an index card, piece of scratch paper, etc. Have students consider the following questions:

  • What strategies did you find effective?

  • How did you decide how to start a problem?

  • What would you not recommend?

Use student responses to see which misconceptions persist before moving on to the next lesson: “Trig Identities, Part 2.”


K20 Center. (n.d.). I Notice, I Wonder. Strategies.

K20 Center. (n.d.). Try It, Talk It, Color It, Check It. Strategies.

International Spy Museum. (2023). Language of espionage.

Spyscape. (2023). The Spyscape glossary of spy terms. SPYSCAPE.