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Algebra 2Algebra 262 views·Updated May 19, 2026·2 pages

Master Trigonometry Identities

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Layla Guthrie@laylaguthrie_aoly

Ready to master trigonometric identities? These powerful relationships between trig... Show more

1
of 2
7. $\cos\theta\cot\theta=\frac{1}{\sin\theta}-\sin\theta$ $\qquad = \frac{1-\sin^2\theta}{\sin\theta} = \frac{\cos^2\theta}{\sin\theta}$ $

Verifying and Simplifying Trigonometric Identities

Trigonometric identities are equations that are true for all values in their domain. The most fundamental identity to remember is sin²θ + cos²θ = 1, which serves as the foundation for many others.

When verifying identities, work with one side of the equation until it matches the other side. For example, in the identity cos⁡θtan⁡θ=sin⁡θ\cos\theta\tan\theta = \sin\theta, we can substitute tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta} to get cos⁡θ⋅sin⁡θcos⁡θ=sin⁡θ\cos\theta \cdot \frac{\sin\theta}{\cos\theta} = \sin\theta, which simplifies to sin⁡θ=sin⁡θ\sin\theta = \sin\theta.

To simplify expressions like tan⁡θcot⁡θ\tan\theta\cot\theta, remember that these are reciprocals, so tan⁡θcot⁡θ=sin⁡θcos⁡θ⋅cos⁡θsin⁡θ=1\tan\theta\cot\theta = \frac{\sin\theta}{\cos\theta} \cdot \frac{\cos\theta}{\sin\theta} = 1. Similarly, sin⁡θcsc⁡θ=1\sin\theta\csc\theta = 1 and cos⁡θsec⁡θ=1\cos\theta\sec\theta = 1 because csc and sec are reciprocals of sin and cos respectively.

Pro Tip: When simplifying complex expressions, try converting everything to sines and cosines first, then look for patterns like sin²θ + cos²θ = 1 or sec²θ - tan²θ = 1 to help simplify further.

2
of 2
7. $\cos\theta\cot\theta=\frac{1}{\sin\theta}-\sin\theta$ $\qquad = \frac{1-\sin^2\theta}{\sin\theta} = \frac{\cos^2\theta}{\sin\theta}$ $

Applying Identities to Find Unknown Values

When you know one trigonometric function and the quadrant of the angle, you can find all other trig functions. This is super useful for solving problems without a calculator!

For example, if sin⁡θ=0.48\sin\theta = 0.48 and θ is in the second quadrant, we can find cos⁡θ\cos\theta using the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1. Substituting gives us (0.48)2+cos⁡2θ=1(0.48)^2 + \cos^2\theta = 1, so cos⁡2θ=1−0.2304=0.7696\cos^2\theta = 1 - 0.2304 = 0.7696. Since θ is in the second quadrant, cos⁡θ\cos\theta is negative, so cos⁡θ=−0.7696≈−0.877\cos\theta = -\sqrt{0.7696} ≈ -0.877. Then we can find tan⁡θ=sin⁡θcos⁡θ=0.48−0.877≈−0.547\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{0.48}{-0.877} ≈ -0.547.

The unit circle is an incredibly helpful tool for visualizing these relationships. It can help you verify identities like sin⁡(θ+π)=−sin⁡θ\sin(\theta + \pi) = -\sin\theta by showing how angles related by π radians appear on opposite sides of the circle.

Remember: In different quadrants, the signs of trig functions change! In the first quadrant, all trig functions are positive; in the second, only sin is positive; in the third, only tan is positive; and in the fourth, only cos is positive. The mnemonic "All Students Take Calculus" can help you remember this pattern.

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Algebra 2Algebra 262 views·Updated May 19, 2026·2 pages

Master Trigonometry Identities

user profile picture
Layla Guthrie@laylaguthrie_aoly

Ready to master trigonometric identities? These powerful relationships between trig functions can make complex problems simple once you understand how they work. Let's break down the key identities and see how to apply them to verify and simplify expressions.

1
of 2
7. $\cos\theta\cot\theta=\frac{1}{\sin\theta}-\sin\theta$ $\qquad = \frac{1-\sin^2\theta}{\sin\theta} = \frac{\cos^2\theta}{\sin\theta}$ $

Sign up to see the content. It's free!

  • Access to all documents
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  • Join milions of students

Verifying and Simplifying Trigonometric Identities

Trigonometric identities are equations that are true for all values in their domain. The most fundamental identity to remember is sin²θ + cos²θ = 1, which serves as the foundation for many others.

When verifying identities, work with one side of the equation until it matches the other side. For example, in the identity cos⁡θtan⁡θ=sin⁡θ\cos\theta\tan\theta = \sin\theta, we can substitute tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta} to get cos⁡θ⋅sin⁡θcos⁡θ=sin⁡θ\cos\theta \cdot \frac{\sin\theta}{\cos\theta} = \sin\theta, which simplifies to sin⁡θ=sin⁡θ\sin\theta = \sin\theta.

To simplify expressions like tan⁡θcot⁡θ\tan\theta\cot\theta, remember that these are reciprocals, so tan⁡θcot⁡θ=sin⁡θcos⁡θ⋅cos⁡θsin⁡θ=1\tan\theta\cot\theta = \frac{\sin\theta}{\cos\theta} \cdot \frac{\cos\theta}{\sin\theta} = 1. Similarly, sin⁡θcsc⁡θ=1\sin\theta\csc\theta = 1 and cos⁡θsec⁡θ=1\cos\theta\sec\theta = 1 because csc and sec are reciprocals of sin and cos respectively.

Pro Tip: When simplifying complex expressions, try converting everything to sines and cosines first, then look for patterns like sin²θ + cos²θ = 1 or sec²θ - tan²θ = 1 to help simplify further.

2
of 2
7. $\cos\theta\cot\theta=\frac{1}{\sin\theta}-\sin\theta$ $\qquad = \frac{1-\sin^2\theta}{\sin\theta} = \frac{\cos^2\theta}{\sin\theta}$ $

Sign up to see the content. It's free!

  • Access to all documents
  • Improve your grades
  • Join milions of students

Applying Identities to Find Unknown Values

When you know one trigonometric function and the quadrant of the angle, you can find all other trig functions. This is super useful for solving problems without a calculator!

For example, if sin⁡θ=0.48\sin\theta = 0.48 and θ is in the second quadrant, we can find cos⁡θ\cos\theta using the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1. Substituting gives us (0.48)2+cos⁡2θ=1(0.48)^2 + \cos^2\theta = 1, so cos⁡2θ=1−0.2304=0.7696\cos^2\theta = 1 - 0.2304 = 0.7696. Since θ is in the second quadrant, cos⁡θ\cos\theta is negative, so cos⁡θ=−0.7696≈−0.877\cos\theta = -\sqrt{0.7696} ≈ -0.877. Then we can find tan⁡θ=sin⁡θcos⁡θ=0.48−0.877≈−0.547\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{0.48}{-0.877} ≈ -0.547.

The unit circle is an incredibly helpful tool for visualizing these relationships. It can help you verify identities like sin⁡(θ+π)=−sin⁡θ\sin(\theta + \pi) = -\sin\theta by showing how angles related by π radians appear on opposite sides of the circle.

Remember: In different quadrants, the signs of trig functions change! In the first quadrant, all trig functions are positive; in the second, only sin is positive; in the third, only tan is positive; and in the fourth, only cos is positive. The mnemonic "All Students Take Calculus" can help you remember this pattern.

We thought you’d never ask...

What is the Knowunity AI companion?

Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Most popular content in Algebra 2

9

Most popular content

9

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.6/5App Store
4.7/5Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan SiOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha KlichAndroid user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

AnnaiOS user