For each expression in Column I, choose the expression from Co...

Question

For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.

cos² x

II

A. sin ^2 x/cos ^2 x

B.1/(sec ^2 x)

C. sin (-x)

D. csc ^2 x-cot ^2 x + sin ^2 x

E. tan x

Accepted Answer

Hey, everyone in this problem, we're asked which of the following expressions is equivalent to tangent squared of X. We have four answer choices. Option A second squared of X plus one. Option BC C can't squared of X minus one. Option C sine squared X divided by C squared X and option D one divided by C can't squared X. All right. So let's start with the expression. We were given tangent squared of X. I recall that tangent is equivalent to sine divided by cosine. And so tangent squared is just going to be sin of X divided by cosine of X all squared. We can use our exponent rules to distribute that squared to both the numerator and denominator. And so this can be written as sine squared of X divided by cosine squared of X. OK. And this answer matches with answer choice C so this looks good. This looks like what the correct answer is going to be. But what we can also do is just to double check the others. OK, to eliminate the others. So if we were to look at option A, we were given C can't squared of X plus one we recall that C can't squared of X is one divided by cosine squared of X. OK. So we have one divided by cosine squared of X plus one. Let's find a common denominator to simplify. So the second term we're gonna multiply numerator and denominator by cosine squared of X. And we get one plus cosine squared of X all divided by cosine squared to X hm. So now we have this cosine squared X in the denominator like we have above but one plus cosine squared of X is not sign squared X. OK. Recall that S score of X who was cosine squared to X is equal to one. OK. So cosine squared of X is going to be equal to one minus sine squared of X. If we want to substitute that into our expression, C squared X plus one becomes one plus one minus sine squared of X divided by cosine squared of X, which is going to be two minus sine squared X divided by cosine squared X. And we know that this is not equal to tangent squared X. OK. We know that we can write it as sine squared X divided by cosine squared X. We're just making sure that none of these other options could also be true. OK. All right. So option A is out for sure, eliminate that. Option B, we can do the exact same thing. We're given that koi can't squared of X minus one. We recall that one plus cotangent squared of X is equal to KC can't square to X. So we can write our expression as one plus cotangent squared of X minus one. OK. Well, this just equals the code tangent squared of X. And we know that code tangent is the reciprocal of tangent. So this does not equal tangent squared of X. OK. It's the reciprocal of what we were given. All right. And then the final option answer D one divided by C can't squared of X. Well, we know that cosine is the reciprocal of C can OK? Or vice versa. C can is the reciprocal of cosine. So this is just gonna give us cosine squared of X. OK. So that one's not correct either. And so that initial simplification we did that is the only option that is equivalent to the expression we were given. So we have that tangent squared of X is equivalent to sine squared of X divided by cosine of squared of X, which is option C thanks everyone for watching. I hope this video helped see you in the next one.

For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.
cos² x

II
A. sin ^2 x/cos ^2 x
B.1/(sec ^2 x)
C. sin (-x)
D. csc ^2 x-cot ^2 x + sin ^2 x
E. tan x

Key Concepts

Pythagorean Identities

Trigonometric Expression Manipulation

Understanding Function Notation and Powers

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