Verify that each equation is an identity.
cos x = (1 - t...

Question

Verify that each equation is an identity.

cos x = (1 - tan² (x/2))/(1 + tan² (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 the sign of two X equals. And then on the right hand side, there's a fraction in the numerator, we have one minus tangent squared X that's divided by C can't squared X. Our answer choices are answer choice. A yes and answer choice B no. All right. So we've got this sign of two multiplied by X. So it looks like we're going to work toward a double angle identity. We recall from previous lessons, the double angle identity with sine is that the sine of two A equals two sine A cosine A. And here, since we've got the sine of two X, we're going to say that our A is equal to X. And then what we're going for is the sine of two X equals two sine X cosine X. And what we're going to do with this one is we're going to work with this right hand side of our given equation and see if we can make it look like the right hand side of our double angle identity. So let's work with our fraction, we have one minus tangent squared X all divided by the C can't squared X. And we can break that into two parts. We can say that that is one divided by C can't squared X minus tangent squared X divided by C can't squared X. And now we recall from previous lessons that the C can't is the reciprocal function of the cosine function. So we recall that the cosine squared X would equal one divided by the C can't squared X. So we can go ahead and substitute in for one divided by C can't squared X. We'll put in cosine squared X and then bring down the minus tangent squared X divided by. And there we have another, if we were to think of the tangent squared X as being multiplied by one that there's that C can't squared X in the denominator. So instead what we can do is we can take that C can't squared X in our denominator. And we can factor that out. It would be essentially we'd have tangent squared X multiplied by one divided by C squared X. And then we can substitute in cosine squared X again. So here we have cosine squared X minus tangent squared X multiplied by cosine squared X. And so we've got cosines and we want to get into in terms of sine and cosine, we're going for our two sine X cosine X and we can get from tangent to sign and cosign, we recall the quotient identity where the tangent squared X would be equal to sine squared X divided by cosine squared X. So we can substitute that in for the tangent squared X which will give us cosine squared X minus sine squared X divided by cosine squared X. That fraction is multiplied still by cosine squared X. But hey, now there's a cosine squared X in both the numerator and the denominator those cancel out. And this is going to simplify to be cosine squared X minus sine squared X. However, that is just not similar enough. It is not the same as two sine X cosine X, we can't get from one to the other. So this here is not an identity, these are not equal. And so we look at our answer choices and this matches with answer choice B well done. We'll catch you on the next one.

Verify that each equation is an identity.
cos x = (1 - tan² (x/2))/(1 + tan² (x/2))

Key Concepts

Trigonometric Identities

Half-Angle Formulas

Algebraic Manipulation of Trigonometric Expressions

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