Graph each expression and use the graph to make a conjecture, ...

Question

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.

(1 - cos 2x)/sin 2x

Accepted Answer

Hello, everyone. We are asked to simplify the given expression. We want to express our answers in terms of X only. We are given one plus the cosine of two X divided by the sine of two X. Our answer choices are a, the tangent of X B, the tangent of X divided by two, see the co tangent of X D, the cotangent of X divided by two. So looking at this, I notice I have double angles. So I'm gonna use some double angle identities. So rewriting our original, we have one plus the cosine of two X divided by the sine of two X. So first, I recall the cosine of two X has the identity that it equals the cosine squared of X minus the sign squared of X. And the double angle identity for the sine of two X is that it equals two sine of X cosine of X. So I'm gonna go back and plug those identities in for the double angles in our original expression. So we will have one plus the cosine squared of X minus the sine squared of X divided by two sine of X cosine of X. So next, I'm gonna rearrange my numerator a little bit, I'm rearranging it. So in parentheses, I have one minus the sign squared of X plus the cosine squared of X divided by two sine of X cosine of X. I do this because I recall that one minus the sine squared of X is equal to the cosine squared of X. So we had that identity, that one minus the sine squared of X will equal the cosine squared of X. So this means I could put cosine squared of X in the numerator here where I had one minus the sine squared of X. And that gives me cosine squared of X plus cosine squared of X divided by two sx cosine of X. So now in my numerator, I have two cosine squares of X. So I'm going to write it that way two cosine squared X and that's divided by two sine X cosine X. And now I wanna reduce, I could reduce the twos. And because cosine squared of X is the same as cosine X multiplied by cosine X, I could reduce the cosine of X in the denominator with one of the cosine of X's in the numerator. This leaves me with the cosine of X divided by the sine of X. And I recognize that cosine of X divided by sin of X is the same as the cotangent of X. So after some factoring and reducing our original expression is equal to the cotangent of X which is answer choice C have a nice day.

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.
(1 - cos 2x)/sin 2x

Key Concepts

Trigonometric Identities

Graphing Trigonometric Functions

Algebraic Verification

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