Let csc x = -3. Find all possible values of (sin x + cos x)/sec x.

Video transcript

Hello, everyone. We are asked to determine the value or values of the given trigonometric expression. If the sine of X equals one third, we are given the CN of X minus the cosine of X divided by the cosine of X. Our answer choices are a 1/8 B negative 1/8 C negative 1/8 positive 1/8 D negative 1/4 positive 14. First noticing that we are given the sign of X and the majority of our expression has to do with the cosine. I'm gonna go find what is the cosine of X. So knowing that the sign of X equals one third and that sin is the ratio of Y divided by R. I can look at the cosine of X as the ratio of X divided by R. And use the Pythagorean theorem to find our missing value X so X squared plus one squared which was Y equals R squared, which is three squared. So X squared plus one equals nine, subtracting one from both sides, X squared equals eight. I'm gonna take the square root of both sides and actually equal plus or minus the square root of eight, which is the same as plus or minus two radical two. So plugging that in I know cosine is plus or minus two radical two divided by R, which is three. Now looking again at our expression, we need to find the C can of X. Also to plug that in thinking through my identities. I recall the sin of X is the same as one divided by the cosine of X. So now that I know the cosine of X, I can say this is one divided by plus or minus two radical two divided by three, which means the C can of X would be plus or minus three divided by two radical two. Let's rationalize that denominator by multiplying top and bottom by the square to two, we will have plus or minus three radical two divided by four. So now we have values for the cosine and the C of X, we're gonna plug this in twice, once with a positive, once with a negative. So we're gonna start with the positive. So again, if we're looking at the CC of X minus the cosine of X divided by the cosine of X starting with positives. So positive three radical two divided by four for our C can minus the cosine, which would be two radical two divided by three divided by the cosine two radical two divided by three. So starting by simplifying my numerator, I'm gonna get them to have a common denominator of 12 I do this by multiplying the first fraction, top and bottom by three. So I'll have nine radical two divided by 12 minus my second fraction to get 3 to 12. I multiplied by four. So I'd have eight radical two divided by 12 in the denominator that's gonna stay as is for now. So two radical two divided by three in the numerator simplifying this, I will have one radical two divided by 12, all of which is divided by two radical two divided by three. I want to rewrite this horizontally because it makes it easier for me to see it. So the square to two divided by 12 divided by two square to two divided by three. And recall when we divide fractions were multiplying by the reciprocal. So the square to two divided by 12 multiplied by three divided by two square root of two, I can cross reduce here. So I'm going to cross, reduce the square to twos out of this problem. Then I'm gonna reduce 3 to 1 and 12 to 4. And I get that this when I multiply across equals 1/8. All right. So we know that this could possibly have a value of 1/8 which is one of our possible answers. It's a choice A and C. So let's see what else we need to do. Continuing on. We're gonna do this exact formula again. But now we're gonna use the negatives. So the C of X minus the cosine of X divided by the cosine of X. So the negatives we'd have negative three radical two divided by four for the C of X minus negative two radical two divided by three for the cosine of X divided by negative two radical two divided by three for the cosine of X. So very similar steps to the previous one, I'm gonna again create a common denominator in my numerators which will be 12 and even the values in the top will be pretty much the same. But the opposite signs will have negative nine radical two divided by 12 and then subtracting a negative is like adding. So I have plus and then we'll have eight radical two divided by 12. And in the denominator, we still have negative two radical two divided by three. Simplifying this, I get negative radical two divided by 12 divided by negative two radical two divided by three rewriting that horizontally I have negative squared at two divided by 12, divided by negative two radical two divided by three. So negative two radical two divided by 12 multiplied by three, divided by negative two radical two cross reducing what I can. I'll reduce the square to twos. I'll reduce the negatives. Three becomes a one, 12 becomes a four. And when I multiply straight across, I again get a positive 1/8. So since both positive and negative give me the value of 1/8 I am comfortable saying that my answer choice. A 1/8 is the answer. Have a nice day.