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.

sec x/csc x + csc x/sec x

Accepted Answer

Hello, everyone. We are asked to simplify the given trigonometric expression. We are given the tangent of X divided by the sine of X plus the cotangent of X divided by the cosine of X. Our answer choices are ac can of X minus Koi can of XB, the C can of X plus the Kosi can of XC two C can of XD two kosi can of X. All right, I am going to rewrite this first. So the tangent of X divided by the sine of X plus the cotangent of X divided by the cosine of X. I'm going to rewrite tangent and cotangent in terms of sign and cosign because I think that'll make this a little simpler recall that the tangent of X is the same as the sine of X divided by the cosine of X, all of which will be divided by the sine of X. And in my second fraction, the co tangent of X is the cosine of X divided by the sine of X, all of which is divided by the cosine of X. Let me see if I can simplify these things at all. So I'm gonna take the first complex fraction and rewrite it as division. So I'll have the sine of X cosine of X divided by the sine of X. And I'm gonna do the same with my second fraction. So in parentheses, the cosine of X divided by the sine of X will then be divided by the cosine of X. I recall that when I work with fractions, I'm multiplying by the reciprocal when I do division. So this would be the sine of X divided by the cosine of X multiplied by one divided by the sine of X. And in my second parentheses, I'd have the cosine of X divided by the sine of X multiplied by one divided by the cosine of X. From here trying to simplify this. I see that I can reduce the sin of X. In my first parentheses, one numerator, one denominator leaving me with one divided by the cosine of X plus. In my second set of parentheses, I could reduce the cosines of X leaving one divided by the sine of X using my reciprocal identities. I recall that the one divided by the cosine of X is the secret of X and one divided by the sine of X is the koi can of X. So this is as simplified as it can get and we have the KC can of X plus the C can of X or the C can of X plus the koi can of X which is answer choice. B, have a nice day.

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Perform each indicated operation and simplify the result so that there are no quotients.
sec x/csc x + csc x/sec x