The half-angle identity
tan A/2 = ± √[(1 - cosA...

Question

The half-angle identity

tan A/2 = ± √[(1 - cosA)/(1 + cos A)]

can be used to find tan 22.5° = √(3 - 2√2), and the half-angle identity

tan A/2 = sin A/(1 + cos A)

can be used to find tan 22.5° = √2 - 1. Show that these answers are the same, without using a calculator. (Hint: If a > 0 and b > 0 and a² = b², then a = b.)

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the tangent of 67.5 degrees. Our answer choices are a, the square root of in parentheses three plus radical two B, the square root of in parentheses three minus two radical two C, the square root of in parentheses three minus radical two D, the square root of in parentheses three plus two radical two. First, I recognize that the tangent of 67.5 degrees will likely lend itself to using a half angle identity where the tangent of A divided by two would be equal to plus or minus the square root of one minus the cosine of A divided by one plus the cosine of A. So in our case, if we say that the tangent of A divided by two is equal to the tangent of 67.5 degrees, this means that A would have to be 135 degrees. So we're gonna use this to find the exact value. So again, we know the tangent of a divided by two or in this case, the tangent of 67.5 degrees is equal to the square root of one minus the cosine of a. So the cosine of 135 degrees divided by one plus the cosine of 135 degrees. And since 67.5 degrees is in quadrant one, we're gonna only worry about the positive value using a unit circle. I'm gonna find the value of the cosine of 135 degrees and then I'm gonna substitute it into my formula. So I'll have one minus and the value of the cosine of 135 degrees is negative radical two divided by two. And that'll be divided by one plus negative radical two divided by two. I now want to simplify what's under the radical. So looking at my numerator, I want to combine these, if I can double negatives are like adding and I'm gonna add one as if it's two divided by two. So I will get two plus radical two divided by two in my numerator. And in my denominator, I'll do the same idea and I'll get two minus radical two divided by two. We continue up in the next column. So the tangent of 67.5 degrees that's equaling the square root. And now with my complex fraction under the radical, I recognize they have the same denominator. This means I can cancel out that denominator to make a simpler fraction. So I'll have the square root of two plus radical two divided by two minus radical two within my fraction. I want to rationalize that denominator. So I'm gonna do so by multiplying both the numerator and the denominator by the conjugate of the denominator. So the conjugate is the same terms but with a different middle sign. So I'll be multiplying both by two plus radical two. And when I simplify that I get the square root of six plus four radical two, I used foil to multiply together my two by no meals. And then I simplified in the denominator, I will get, let's see, two, multiplied by two. So four and then the square root of two multiplied by the square root of two is two. So this should be four minus two, which is two. So right now in my simplification, I have the square root of six plus four radical two, all of which is divided by two. I think I can reduce this one more time though rewriting the tangent of 67.5 degrees. And since each term under my radical is divisible by two, I'm gonna reduce it by two. So I'll get the square root of three plus two radical two. And that is our exact value. And it matches answer choice D which is the square root of three plus two. Radical two have a nice day.

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