Simplify each expression.
√[(1 + cos 165°)/(1 ...

Question

Simplify each expression.

√[(1 + cos 165°)/(1 - cos 165°)]

Accepted Answer

Hello, everyone. We are asked to simplify the given trigonometric expression, we are given the square root of the fraction one plus the cosine of 105 degrees divided by one minus the cosine of 105 degrees. Our answer choices are a, the tangent of 52.5 degrees. B the contingent of 52.5 degrees. See the co tangent of 105 degrees, the, the tangent of 105 degrees. So looking back at our original expression, I noticed that it looks similar to the half angle identity for a tangent where the tangent of A divided by two equals plus or minus the square root of one minus the cosine of A divided by one plus the cosine of A. However, in our case, we have under the radical, the addition in the numerator subtraction in the denominator. So that strikes me as if it's a reciprocal. So I'm gonna use a as 105 degrees. We're gonna say the tangent of 105 degrees divided by two, which means we're working with the tangent at the moment of 52.5 degrees. And that would have to equal the square root of one minus the cosine of 105 degrees divided by one plus the cosine of 105 degrees. Since I want to flip this fraction, I wanna take the inverse or the reciprocal of both. So if I do one divided by the tangent of 52.5 degrees, see that's equal to one divided by the square root of one minus the cosine of 105 degrees divided by one plus the cosine of 105 degrees. I recall my reciprocal identities tell me that one divided by the tangent of an angle is the cotangent of that angle. So this would be the cotangent of 52.5 degrees and that would equal the reciprocal of our other fraction which will still work out to be the square root and it will be of one plus the cosine of 105 degrees divided by one minus the cosine of 105 degrees. This now matches our original expression. So we could say that this is equal to the cotangent of 52.5 degrees, which is answer choice B have a nice day.

Simplify each expression.
√[(1 + cos 165°)/(1 - cos 165°)]

Key Concepts

Trigonometric Identities

Cosine Function Properties

Rationalizing Expressions

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