Using the quotient identities to prove trig identities
About this video
The video proves that tan(theta) x cot(theta) = 1 by rewriting tangent as sin(theta)/cos(theta) and cotangent as cos(theta)/sin(theta). Multiplying gives the same product on top and bottom, which equals 1, showing that breaking things into sine and cosine is a good first step in proofs.
You'll be able to
- write tangent and cotangent in terms of sine and cosine.
- prove a simple trig identity by substituting definitions and simplifying.
- explain why rewriting in sine and cosine is a good starting strategy.
Key words
- trigonometric identity
- An equation with trig functions that is true for every allowed angle.
- tangent
- tan(theta) = sin(theta) / cos(theta).
- cotangent
- cot(theta) = cos(theta) / sin(theta), the reciprocal of tangent.
- reciprocal
- One divided by a number; flipping a fraction upside down.
Transcript
- [Instructor] What we're gonna do in this video is get a little practice proving trigonometric identities, and we will do that by proving out that tangent of theta times cotangent, not cosine, cotangent of theta, let me put parentheses around this to make this clear, is equal to one.
So like always, pause this video and see if you can prove this out. Alright, now let's break this down a little bit. So the key here is to just go straight to the definitions of tangent and cotangent in terms of sine and cosine. We know that tangent of theta is equal to sine of theta over cosine of theta, and we know that cotangent of theta is just the reciprocal of that, cotangent of theta is equal to cosine of theta over sine of theta.
So if we just rewrite the left-hand side of this expression right over here, we can rewrite tangent of theta as sine of, I'll use the parentheses, sine of theta over cosine of theta, and then we can rewrite cotangent of theta as, so we're multiplying this times cosine of theta over sine of theta.
Well, we can immediately see that in our numerator we have sine of theta times cosine of theta, and our denominator, we have sine of theta times cosine of theta. Or you could even think about it that sine of theta divided by sine of theta is one over one, and cosine of theta over cosine of theta or cosine over cosign is also going to be one for any theta.
So we end up that this is just going to be equal to one. So this is just an example of one proof that you can do, but that's always a good place to start. If something can be broken down using definitions like this, try to do that. See what happens when you break it down that way.
Now, there's gonna be other techniques in the future, but this is always a great place to start.
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