Use identities to write each expression in terms of sin θ and ...

Question

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.

csc² θ + sec² θ

Accepted Answer

Hello, everyone. We are asked to write the following trigonometric expression. In terms of sine and cosine, we will express our answer in terms of X only. We are given the tangent squared of X plus the cotangent squared of X. Our answer choices are a sine to the fourth power of X plus cosine squared X divided by cosine squared X multiplied by sine squared X B sine raised to the fourth power X plus cosine to the fourth power X divided by cosine squared X sine squared X C one divided by cosine squared X sine squared X D sine squared X plus cosine to the fourth X divided by cosine squared X sine squared X. All right, we have a lot going on in those answers. So let's see what happens when we start putting this in terms of sign and cosign so we have tangent squared X plus cotangent squared X. So first tangent tangent of X is the sine of X divided by the cosine of X. So I'm gonna put this in parentheses. So this will be the sine of X divided by the cosine of X and the parentheses will be squared because our tangent is squared. We're gonna add to that the cotangent squared of X. So cotangent of X is the cosine of X divided by the sine of X. And again, because this is squared, I'm gonna put a square outside of my parentheses. So now I'm gonna square what's in the parentheses and see where I need to go from there. So I will have sine squared X divided by cosine squared X plus cosine squared X divided by sine squared X. Recall in order to add fractions, they must have a common denominator. So my common denominator here is going to be sine squared X cosine squared X or vice versa. So to do that, I'm gonna multiply my first fraction, both numerator and denominator by sine squared X. And to make my common denominator, my second fraction will be multiplied by cosine squared X for both numerator and denominator. And when I do so sine squared X multiplied by sine squared X will give me sine to the fourth X and that will be divided by cosine squared X sine squared X for my second fraction, I'll have cosine squared X multiplied by cosine squared X which is cosine to the fourth X and that'll be divided by cosine squared X sine squared X. So now that I have a common denominator of cosine squared X sine squared X, I'm gonna write that as my denominator and combine my name readers best I can, they are not like terms. So all I can do is write them as a single numerator with a plus sign between them. And I get sine to the fourth power X plus cosine to the fourth power X. All of which is divided by cosine squared X sine squared X. There is nothing else that can be simplified. And so that is our expression and it matches answer choice B have a nice day.

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.
csc² θ + sec² θ

Key Concepts

Cosecant and Secant Functions

Pythagorean Identities

Simplification of Trigonometric Expressions

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