Verify that each equation is an identity.
cot² (x/2) = (...

Question

Verify that each equation is an identity.

cot² (x/2) = (1 + cos x)²/(sin² x)

Accepted Answer

Welcome back. I am so glad you're here. We are asked is the given equation an identity. And our given equation is the cotangent squared of theta divided by two is equal to. Then we have a fraction in the numerator. We have the cosine squared of theta plus the cosine of theta plus two all divided by sine squared. Theta. Our answer choices are answer choice. A yes and answer choice B no. OK. So we wanna find something to work with here. And we can see that on the left hand side where we have the cotangent squared of theta divided by two, this could lead us to a half angle identity. So we want to look at the half angle identities and we don't have one for cotangent, but we do have one for tangent, which we recall from previous lessons is the reciprocal function of cotangent. So let's start off by using the half angle identity that the tangent of A divided by two equals. Now, looking at the right hand side of what we've got, we know that we're dealing with a reciprocal function. So for our half angle identity, we would want to have what's in the denominator or something similar for our given equation. So here we can use in the numerator, our tangent of A divided by two equals the sign of A divided by. And then our denominator that goes with this one is one plus the cosine of a, which is similar to our numerator, similar to our numerator in our given equation. So we're going to start working with this one. We can see that our a in our half angle identity can be substituted for theta. So we can say anywhere in here we have an A, we're going to put in theta and we can rewrite this half angle identity as the tangent of theta divided by two equals the sine of theta divided by one plus the cosine of theta. And as we said earlier, tangent is the reciprocal function of cotangent. So we can say that the cotangent of this same angle, it's still theta divided by two equals. Now the reciprocal of the right hand side. So we'll have in the numerator one plus the cosine of theta and that's divided by the sine of theta. Now, the only thing we need to do now to get it to look like the original for the left hand side is we need to square it. So we'll have a cotangent squared. So we're going to square both sides of our equation that we're doing the work on. And then that will simplify to be on the left hand side, the cotangent squared of theta divided by two good. That's looking good. Now, on the right hand side, I'm just gonna take it a few steps at a time. We're squaring both our numerator and our denominator. So in the numerator, we have the quantity of one plus cosine theta squared, we're going to need to foil that divided by, in the denominator, sine, theta squared is sine squared theta. So that denominator is looking good, let's just foil out our numerator. So we'll take that aside and foil this out. So we have the quantity of one plus cosine theta multiplied by the quantity of one plus cosine theta multiplying our firsts, one multiplied by one is one multiplying our outsides plus our insides multiplied together is going to be plus two cosine theta and multiplying our lasts cosine theta multiplied by itself is going to be plus cosine squared theta. So we can put that in our, in our exponential order from greatest to least. So we would have, let's write this whole thing. We have the cotangent squared of theta divided by two that all matches equals in our numerator. We'll start with cosine squared theta plus two, cosine theta plus one. And that's all divided by sine squared theta. Well, we know the left hand side matches, we know our denominator matches. How about our numerator. Well, this one doesn't match, we have cosine squared theta, but then we have plus two cosine theta instead of plus just cosine data and then plus one instead of plus two. So this one is not an identity and we look at our answer choices and this matches with answer choice B so close. Well done, we'll catch you on the next one.

Verify that each equation is an identity.
cot² (x/2) = (1 + cos x)²/(sin² x)

Key Concepts

Trigonometric Identities

Half-Angle Formulas

Algebraic Manipulation of Trigonometric Expressions

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