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

Question

Verify that each equation is an identity.

(sin 2x)/(2sin x) = cos² (x/2) - sin² (x/2)

Accepted Answer

Welcome back. I am so glad you're here. We are asked is the given equation an identity. Our given equation is on the left hand side, we have a fraction in the numerator. It's two sine theta and that's divided by the sine of two theta that's equal to on the right. We also have a fraction in the numerator. It's one and that's divided by, in the denominator two cosine squared of they are divided by two minus one. Our answer choices are answer choice A yes and answer choice B no. All right. So we want to look for an in somewhere we can get in here to use some trigonometric identities. And then simplify. And here our in looks like it's going to be on the left hand side in the denominator where we have the sign of two, theta, our theta is being multiplied by two. So perhaps we can use a double angle identity here. Since it's the sign of two theta, we recall from previous lessons, the double angle identity where the sign of two A equals two sine A cosine A. And this can be really helpful for this particular double angle identity we'll set a equal to theta and now we can substitute in for the sine of two theta. So let's bring this down in the numerator. We have two sine theta and that's divided by now instead of sine of two theta, we'll put what the sine of two theta is equal to which is two sine theta cosine theta and we bring down the rest that equals on the right one divided by two cosine squared theta divided by two minus one. And look at that on the left, we have two sine theta in both the numerator and the denominator. Those cancel out, those are ones. And now we have one divided by the cosine of theta. On the left equals on the right one divided by two cosine squared. They get it divided by two minus one. And now we have two fractions equal to each other. Well, the reciprocals are going to be equal to each other as well. Since we have one in the numerators, the reciprocals are great because then we get rid of the fractions. So this will be on the left, the cosine of theta equals on the right two cosine squared theta divided by two minus one. And this actually looks a lot like another double angle identity. Recalling from those same previous lessons and tables in the textbooks, we know that the cosine of two A equals two cosine squared A minus one. So that looks very similar, but it's a little bit different. But we can see on the left, we just have cosine of theta, not the cosine of two something, but on the right where we have two cosine squared A, our A over on the right hand side would be theta divided by two. So if we set a equal to theta divided by two, on the left hand side, plugging in theta divided by two and multiplying that by two. well, that gets us right back to theta. Let me show you what I mean. So if our A is theta divided by two, we have the cosine of two multiplied by theta divided by two equals two cosine squared of theta divided by two minus one on the left, those twos cancel out that gets us right back to where we started. And so yes, this one is a identity and identity. We look at our answer choices and that matches with answer choice. A well done. We'll catch you on the next one.

Verify that each equation is an identity.
(sin 2x)/(2sin x) = cos² (x/2) - sin² (x/2)

Key Concepts

Trigonometric Identities

Double-Angle Formulas

Half-Angle Formulas

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