Perform each indicated operation and simplify the result so th...

Question

Perform each indicated operation and simplify the result so that there are no quotients.

1/( sin α - 1) - 1/(sin α + 1)

Accepted Answer

Welcome back. I am so glad you're here. We are asked to simplify the given trigonometric expression. Our given trigonometric expression is one divided by the cosine of X minus one minus one divided by cosine X plus one. Our answer choices are answer choice A two K can't squared X answer choice B negative two KC can't squared X answer choice C negative two K can't X co tangent X and choice D two K can X cotangent X. All right. So we've got two fractions being subtracted. So the first thing we want to do is find a common denominator so that we can subtract those fractions. One denominator is cosine X minus one and one denominator is cosine X plus one. So the common denominator is going to be cosine X minus one multiplied by the quantity of cosine X plus one. So taking our first fraction which has the cosine X minus one and multiplying both the numerator and the denominator by cosine X plus one, which is essentially one. So we're just multiplying it by one and then taking our second fraction one divided by cosine of X plus one and multiplying both the numerator and the denominator of that one by cosine X minus one. Again, multiplying it by essentially one. Now, we do have that common denominator that we wanted quantity of cosine X minus one multiplied by the quantity of cosine X plus one. And then for that first fraction, one multiplied by cosine X plus one. That's just going to be cosine X plus one, then minus from the second fraction one multiplied by the cosine of X minus one going to put cosine X minus one still in parentheses. So I distribute that negative to both cosine X and the minus one. So let's do that on the next step, we've got cosine X plus one minus cosine X and then minus negative one, which is plus one all in our numerator. And now we can foil out our denominator. So for our first cosine X multiplied by cosine X, that's going to be cosine squared X, our outsides cosine X multiplied by positive one added to our insides, negative one multiplied by cosine X. Those cancel out and now our last negative one multiplied by positive one. That's going to be minus one. Now simplifying in our numerator, we have cosine X minus cosine X those cancel out and then we just have one plus one which is two all divided by cosine squared X minus one. Now cosine squared X minus one looks kind of like our Pythagorean identity. We recall from previous lessons, the Pythagorean identity of sine squared, theta plus cosine squared, theta equals one. We have parts of that. If we subtract one from both sides and we subtract sine squared data from both sides. This would give us cosine squared theta minus one equals a negative sine squared theta. And the left sign the left side if theta is equal to X looks just like our denominator. So because they're equal, we can put in a negative sine squared X for our denominator. So let's do that. We have two divided by a negative sine squared X. Now we recall from previous lessons that sine squared is going to be the reciprocal function of CC can't squared. So if we have, let's say if we have cos can't squared of, we'll say theta that's equal to one divided by sine squared theta. So again, if theta is X, we have our sine squared in the denominator, that means our CC can't, can come up to the numerator and we'll bring that negative up to the numerator as well. So we have a negative two and then instead of having one divided by sine squared X, we now have cos can squared X and negative two KC can't squared X is our simplified expression. We look at our answer choices and that matches with answer choice B well done. We'll catch you on the next one.

Perform each indicated operation and simplify the result so that there are no quotients.
1/( sin α - 1) - 1/(sin α + 1)

Key Concepts

Trigonometric Functions

Common Denominator

Simplification of Expressions

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