Verify that each equation is an identity. See Example 4.

Question

tan(x - y) - tan(y - x) = 2(tan x - tan y)/(1 + tan x tan y)

Accepted Answer

Hello, everyone. We are asked if the given equation is an identity, we are given the tangent of X minus Y plus two tangent of Y minus X equals the tangent of Y minus X divided by one plus the tangent of X tangent of Y. Our answer choices are a yes B no for our purposes here, when we're asked if this is an identity we want to know does the left hand side equal the right hand side through any means. So I think I will be using the difference of angles for a tangent identity as part of this. And that identity states that the tangent of X minus Y equals the tangent of X minus the tangent of Y divided by one plus the tangent of X multiplied by the tangent of Y. And if you notice the second part of our equation on the left hand side is asking us more about the tangent of Y minus X. Good news is we can use the same format. The only thing that changes is the order of our letters. So we'd have tangent of Y minus the tangent of X divided by one plus the tangent of Y tangent of X. So now that I have those values, I'm going to plug this in and try to simplify the left hand side of this equation to see if I can get it to match the right hand side. So the tangent of X minus Y will be rewritten as the tangent of X minus the tangent of Y divided by one plus the tangent of X multiplied by the tangent of Y plus two, which will be multiplied by the fraction, the tangent of Y minus the tangent of X divided by one plus the tangent of why tangent of X. So I think I'm gonna multiply the number two into the numerator of the second fraction. So that way I have a common denominator and I can combine my fractions. So I have the tangent of X minus the tangent of Y divided by one plus the tangent of X tangent of Y and then plus two. And I'm distributing that into the parentheses, tangent of Y minus two tangent of X divided by one plus. And I'm gonna rearrange the tangent of Y tangent of X here. So that is in tangent of X multiplied by the tangent of Y. So they match it exactly. And I can do that because order doesn't matter when we're multiplying writing these is one fraction, my denominator will be one plus the tangent of X tangent of Y, my numerator, I'm gonna combine my leg terms. So starting with the Ys, I have a positive two tangent of Y and a negative one tangent of Y which will give me a positive one tangent of Y. And then I have a negative two tangent of X and a positive one tangent of X to leave me with a negative tangent of X. And now comparing that to the right hand side, it matches identically. So this is an identity and it is answer choice. A yes. Have a nice day.

Verify that each equation is an identity. See Example 4.
tan(x - y) - tan(y - x) = 2(tan x - tan y)/(1 + tan x tan y)

Key Concepts

Trigonometric Identities

Tangent Function and Its Properties

Algebraic Manipulation in Trigonometry

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