Verify that each equation is an identity.
tan (θ/2) = cs...

Question

Verify that each equation is an identity.

tan (θ/2) = csc θ - cot θ

Accepted Answer

Welcome back. I am so glad you're here. We are asked is the given equation an identity. Our given equation is the cotangent of X divided by two equals. Then we have a fraction in the numerator. We have the cosecant of X plus the cotangent of X. And that's all divided by two. Our answer choices are answer choice. A yes and answer. Choice B no. All right. So we've got this equation and we want to use trigonometric identities and see if we can work with those identities to get them to look just like this equation here, we can see on the left hand side, we have the cotangent of X divided by two. So since we're dividing by two, a good place to start is with a half angle identity, the half angle identity we're going to use, we don't have one for cotangent, but we do for its reciprocal function tangent. So we know that the tangent of a divided by two from previous lessons. And from those tables in the textbook is equal to, we have several traces with this one. But since we're dealing with, we don't have square ruts and we've got cos can and cotangent, we can use that the tangent of A divided by two is equal to the sine of A, divided by one plus the cosine of A because we can, we can get from sine and cosine to Kant and cotangent. So now we're going to work with this half angle identity and see if we can get it looking like our given equation. So we want to get from tangent back to cotangent. We know they're reciprocal functions. So we can take this and we know that the cotangent of that same angle A divided by two would be equal to the reciprocal of what the tangent of A divided by two would be. So we'll take our denominator from the tangent 11 plus the cosine of A and put that in the numerator one plus cosine A divided by our numerator from the tangent one that's going to be the sign of A in our denominator. Now, another thing that we can do, we can identify that this is very close to what we have. But here we're dealing with a, there in our given, we're dealing with X. So we can set our A equal to X. So now we can rewrite this. We've got the cotangent of X divided by two equals one plus the cosine of X all divided by the sine of X. So now it's very, it's, it's exactly what we want on the left hand side, but we're not quite there yet. On the right. What we can do, we know that one plus the cosine of X divided by the sine of X is the same as one divided by the sine of X plus the cosine of X divided by the sine of X. So that's equal to the cotangent of X divided by two. And now we recall from previous lessons that cos can't of, we'll just stick with X. Here is the reciprocal function of the sine function. So the Cosan of X is equal to one divided by the sine of X. So we can use substitution there for one divided by the sine of X. We can put in the cosecant of X. So our cotangent of X divided by two is equal to instead of one divided by sine of X, the cosecant of X. Now, very similarly, we recall from previous lessons, the quotient identity that the cotangent of X is equal to the cosine of X divided by the sine of X. So we can again use substitution, we've got the cosine of X divided by the sine of X. And we can put in the cotangent of X. So we'll add the cotangent of X to that cosecant of X. Now, that's about as close as we're going to get with this one to our given equation. We have the co C can of X plus the cotangent of X. But unlike our given equation, it's not divided by two, our given equation was dividing it by two. It's not the same. So this is not an identity and we look at our answer choices and that matches with answer choice B it was very close, but not this one. Well done. We'll catch you on the next one.

Verify that each equation is an identity.
tan (θ/2) = csc θ - cot θ

Key Concepts

Trigonometric Identities

Half-Angle Formulas

Cosecant and Cotangent Functions

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