Verify that each equation is an identity.
sin θ/(1 - cos...

Question

Verify that each equation is an identity.

sin θ/(1 - cos θ) - sin θ cos θ/( 1 + cos θ) = csc θ (1 + cos² θ)

Accepted Answer

Hey, everyone in this problem, we're asked if the given equation is an identity. And when it says an identity, what that means is it's true for every X value. Now, the equation we have cosine of X divided by one plus sine of X plus sine of X multiplied by cosine of X divided by one minus sine of X is equal to cosine of X. And we're given two answer choices. Option A option B no. So what we're gonna do is start with the left side of this equation. We're gonna simplify it as much as we can and we're gonna see if we can get it equal to the right hand side of this equation. On the left hand side, we have cosine of X divided by one plus sin of X plus sign of books, all of a divided by one minus S. Now, in order to start simplifying this, let's find a common denominator. OK. So the first fraction is gonna need to be multiplied by one minus sine of X. This gives us cosine of X multiplied by one minus sine of X in the numerator. The second term will need to be multiplied by one plus sin of X. So in the numerator, we get sin of X cosine of X multiplied by one plus sine of X. And all of this is divided by that common denominator, one plus sin of X multiplied by one minus sine effects. Now, we have our common denominators. Let's go ahead and expand in the numerator and we're gonna do the same in the denominator as well. So we end up with cosign of X minus sign of X cosine of X plus sign of X cosine of X plus sine squared X cosine of X. Yeah. And we're writing every single term out here recommend doing that. So you don't mix up any terms or miss any terms. We've got our four terms in the numerator and in the denominator, we're gonna have the same. So we're gonna have one what sign of X minus sin of X minus sine squared, we can simplify quite a bit here. So in the numerator, we have negative sin X cos X plus sin X cos X those are gonna add up to zero in the denominator, Sinex minus Sinex that adds up to zero. And so we're left with cosine of X plus sign squared effect cosine of X all divided by one minus sine squared LX. Let's continue to simplify in the numerator. We have this common factor of cosine. So we're gonna factor that out cosine of X multiplied by one plus sine squared. Of X all divided by, in the denominator, we have one minus cosine squared of X or sorry, one minus sine squared of X. And recall that sine squared of X plus cosine squared of X is equal to one. If we were to rearrange that one minus sine squared of X will just be equal to cosine of X squared. So we can write the denominator of cosine squared of X. We have cosine of X in the numerator, cosine squared of X in the denominator. So that's gonna leave us with cosine X term in the denominator, we get one plus sine squared X divided by cosine of X. Now we're getting pretty down to the simplified version. This isn't looking like the right hand side. We could go one step further if we wanted, we know that sine squared of X is going to be equal to one minus cosine squared of X. So we could put that in the numerator and we have one plus sine squared of X again, sine squared of X, one minus cosine squared of X, all of this divided by cosine of X. And what we can see is that we cannot simplify this and get this down to cosine of X. And we're gonna have two minus cosine squared of X divided by cosine of X. And this is just not equal to that right hand side. So because these two equations are not equal to each other or the two sides of our equation. Sorry are not equal to each other. That means that this is not an identity. And so the answer here is going to be option B No, thanks everyone for watching. I hope this video helped you in the next one.

Verify that each equation is an identity.
sin θ/(1 - cos θ) - sin θ cos θ/( 1 + cos θ) = csc θ (1 + cos² θ)

Key Concepts

Trigonometric Identities

Reciprocal Functions

Algebraic Manipulation

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