Verify that each equation is an identity.
(1 + sin θ)/(1...

Question

Verify that each equation is an identity.

(1 + sin θ)/(1 - sin θ) - (1 - sin θ)/( 1 + sin θ) = 4 tan θ sec θ

Accepted Answer

Hello, everyone we are asked is the given equation and identity. We are given one plus the cosine of X divided by one minus the cosine of X minus one minus the cosine of X divided by one plus the cosine of X equals four cotangent of XKC can of X. Our answer choices are a yes B no, I believe I will start out on the left hand side. And for this to be an identity, the left hand side must ultimately match the right hand side. So on the left hand side, we have one plus the cosine of X divided by one minus the cosine of X minus one minus the cosine of X divided by one plus the cosine of X. The first thing I want to do here is create a common denominator. My common denominator here is going to be one plus the cosine of X multiplied by one minus the cosine of X. So I will be multiplying my second fraction top and bottom by the binomial one minus the cosine of X. And I will be multiplying the other fraction by one plus the cosine of X for both the numerator and the denominator. And when I do in my first fraction, I will have one plus the cosine of X squared divided by our denominator one plus the cosine of X multiplied by one minus the cosine of X. In my second fraction, my numerator is one minus the cosine of X squared divided by our common denominator one plus the cosine of X multiplied by one minus the cosine of X. I'm going to multiply out my common denominator so that I have the difference of squares because right now we are working with the product of a sum in a difference. So we have one minus cosine squared X as our new denominator. In the same step, I'm going to multiply out the squares that were in my numerator. So for one plus the cosine of X squared, I recall that the shortcut is to square the first term score. The last term take the product of the two terms and double it. So from our first fraction, we will get one plus cosine squared X plus two cosine X and then we have minus and I'm going to put this in parentheses until I get to distribute the negative in and I will have one plus the cosine squared X minus two cosine X. So in my next step, I'll distribute the negative sign into that second set of parentheses. So that way it can be subtracted and I'll have one plus the cosine squared of X plus two cosine of X minus one minus cosine squared X plus two cosine X, all of which is divided by one minus the cosine squared of X. Next, I'm going to combine my like terms in my numerator. I think that'll simplify a lot of what we're looking at here. So starting with the one, I have a positive one and a negative one. So those will cancel out, I have a positive cosine squared X and a negative cosine squared X. So those will cancel out. And I'm left in my numerator with four cosine X divided by one minus the cosine squared of X. From here, I recall that there's a Pythagorean identity that states that one minus the cosine squared of X is equal to the sine squared of X. So I'm gonna put sign squared of X in my denominator and I now have four cosine X divided by sign squared X. I'm gonna split this up into a couple different factors. So when I do so I can break this down to four multiplied by the cosine of X divided by the sine of X multiplied by one divided by the sine of X. So I factored my fraction into three different factors. And now let's rewrite that. So the four stays a four, the cosine of X divided by the sine of X. That identity tells us that is the cotangent of X and one divided by the sine of X is the cosecant of X. So in the end, I did get the left hand side to match the right hand side. So this is answer choice. A yes, it is an identity. Have a nice day.

Verify that each equation is an identity.
(1 + sin θ)/(1 - sin θ) - (1 - sin θ)/( 1 + sin θ) = 4 tan θ sec θ

Key Concepts

Trigonometric Identities

Simplifying Expressions

Tangent and Secant Functions

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