How to Prove & Derive Trigonometric Identities

How to Prove & Derive Trigonometric Identities
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  • 0:01 Trigonometric Identities
  • 0:36 The Tangent
  • 2:25 The Double-Angle Identities
  • 3:55 The Half-Angle Identities
  • 6:36 Lesson Summary
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Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will learn how some of our trigonometric identities are derived. You will also see how some identities naturally lead to the others.

Trigonometric Identities

In trigonometry, we have a bunch of trigonometry identities, or true statements about trig functions. Think of these as definitions if you will. They tell you how to describe certain trig functions in other terms.

We use our trigonometry identities to help us simplify more complicated trig problems and prove other trig statements. What's really neat about some of our identities is that we can easily prove them from the other identities. So, if you ever forget some of them, you could derive them yourself if you remember the proofs that you're about to see. Are you ready to begin? Get your thinking cap on!

The Tangent

The first one we are going to see is the tangent function. Remember that you've already learned that the tangent function also happens to be the sine function divided by the cosine function. How did they come up with this? We can easily derive this using our definitions for each of those functions. Make sure your thinking cap is still on, as this requires a bit of thinking.

First, our definitions - we know that our sine function is defined as opposite over hypotenuse, our cosine function is adjacent over hypotenuse, and our tangent function is opposite over adjacent. Recall that these definitions are based on the right triangle where the hypotenuse is the hypotenuse side, the adjacent is the side closest to the angle, and the opposite is the side opposite to the angle.

We are going to use these definitions to show how we can go from sine over cosine to the tangent function. We begin with our sine/cosine. We then insert our definitions. We get (opposite/hypotenuse) / (adjacent/hypotenuse).

Using our knowledge of dividing fractions, we turn this into a multiplication problem by flipping the bottom fraction. We get (opposite/hypotenuse) * (hypotenuse/adjacent). Now, we can go ahead and cancel or simplify what we can. We see a hypotenuse in the numerator and denominator. We can go ahead and cancel these.

What are we left with? We are left with opposite/adjacent. Which function does this define? Why, isn't it the tangent function? And there we have it; we have derived the tangent function from sine/cosine. The whole process looks like this:

sine/cosine = (opposite/hypotenuse) / (adjacent/hypotenuse) = (opposite/hypotenuse) * (hypotenuse/adjacent) = opposite/adjacent = tangent.

Pretty cool, huh?

The Double-Angle Identities

Now, let's look at something slightly more complicated, but not more difficult. We're going to derive our double-angle identities from our sum and difference identities. Recall that our double-angle identities are these:

derive identities

And our sum identities are these:

derive identities

The difference identities are the same as the sum identities except that the signs are opposite. What this means is that where you see a plus sign now, you will see a minus sign, and where you see a minus sign now, you will see a plus sign. If you don't see a sign in front of something, it stays the same.

For this part of the lesson, we are only going to look at the sum identities that you see. We can derive our double-angle identities from our sum identities by simply setting the angles alpha and beta equal to each other. If alpha and beta were both x, then alpha plus beta will become 2x. If we plug in x for both alpha and beta, we will get these for our sum identities:

derive identities

All we did was plug in x for both alpha and beta and then simplified our expressions. We applied our algebra skills to combine like terms. Do you recognize the formulas that we ended up with? Why, aren't they our double-angle identities?

Yes, they are indeed! If you ever forget your double-angle identities, but you remember your sum identities, then you can easily find the double-angle identities by simply setting both angles in the sum identities to the same value.

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