Verify that each equation is an identity.
sec² α - 1 = (...

Question

Verify that each equation is an identity.

sec² α - 1 = (sec 2α - 1)/(sec 2α + 1)

Accepted Answer

Hey, everyone in this problem, we're asked if the given equation is an identity. And when we're asked if something is an identity, what it means is, does this equation hold true for every X value? And the equation we're given is C cant of eight X minus one divided by C can of eight X plus one is equal to tangent squared of four X. We have two answer choices. Yes or no. We're gonna start with the left hand side. So we have C can of eight X minus one divided by C can of eight X plus one. And we're gonna try to simplify this and write it like the right hand side. Now C can't recall is one divided by cosine. So let's start there and we can write this as one divided by cosine of A X minus one, divided by one divided by cosine ma X plus one. Now, in the numerator and denominator, we're gonna go ahead and find a common denominator for each so that we can simplify our fractions. And when we do that, we have one minus cosine of eight X divided by cosign of eight X all divided by one plus cosine of eight X divided by cosine of eight X. OK. So we found this common denominator of cosine of eight X in both the numerator and denominator. And we've just rewritten one as cosine of eight X divided by cosine of eight X. OK. And the numerator with the minus in the denominator with the plus. Now, both of our numerator and denominator have this cosine of eight X in their denomination. That term is going to divide out and we're just left with one minus cosine of eight X divided by one plus co-signed update. OK. So we started with C camp. We've rewritten this expression as cosign of A X and now we need to use Stop to help us. We have eight X on this expression that we have right now inside of our trig functions. On the right hand side, it has four X. OK? So we need to take that angle and half it divided by two. When we have something like that that we wanna do, we wanna think about using half angle formulas. OK? Now what half angle formula is gonna allow us to get, we recall the tangent of theta divided by two is equal to the sign. OK. So we can just write this as plus or minus the square root of one minus cosine of theta divided by one plus cosign of the All right. So our expression is the stuff inside the square root. So if we square both sides of this equation. We have tangent squared of theta divided by two is going to be equal to. And now the issue with the sign kind of disappears. OK, we're squaring it. So whether we had a positive or a negative when we square it, it becomes positive. So we don't need to worry about that anymore. And we have one minus cosine of beta divided by one plus cosine of the, which looks exactly like the expression we worked at. And the only difference is that we have eight X instead of theta. So what we're gonna do is we're gonna let the be equal to eight X. And then we're gonna use this expression. If theta is equal to eight X, then this entire expression one minus cosine of eight X divided by one plus cosine of eight X is going to be equivalent to tangent squared of that value eight X divided by two. This is just equal to tangent squared of four X, which is exactly what we had on the right hand side. And so the left hand side of this equation, we can rearrange and right as the right hand side, that means both sides are equal, no matter what the X value is. And so this is an identity. If we go back to our answer choices, the correct answer is therefore going to be option A yes. Thanks everyone for watching. I hope this video helped you in the next one.

Verify that each equation is an identity.
sec² α - 1 = (sec 2α - 1)/(sec 2α + 1)

Key Concepts

Trigonometric Identities

Pythagorean Identity

Double-Angle Formulas

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