Solve each equation for exact solutions.
cos⁻¹ x + tan⁻¹...

Question

Solve each equation for exact solutions.

cos⁻¹ x + tan⁻¹ x = π/2

Accepted Answer

Welcome back. I am so glad you're here. We are asked to find the exact solution of the equation. Our given equation is the inverse sine of X plus the inverse C can of X equals pi divided by two. Our answer choices are answer choice. A one answer, choice B negative one answer choice C negative one and one and answer choice D no solution. All right. So we have an inverse trigonometric function plus an inverse trigonometric function equals pi divided by two. And what we're trying to solve for is X, what's the value of X where this is true? So we're going to go through a series of remembering identities and using substitution in order to solve for X. I'm going to start off by subtracting the inverse sequent of X from both sides of the equation. So that this will read the inverse sine of X equals pi divided by two minus the inverse sequent of X. And now just kind of focusing on this right hand side for a moment. I'm going to recall from previous lessons on cof functions that if we have the inverse C can of a function. And we add its coun which is going to be Kan, excuse me, inverse Kant, that's going to equal 90 degrees or pi divided by two. They are co functions. So if we were to then subtract the inverse C can of X from both sides, we would have on the left, the inverse cosecant of X equals on the right pi divided by two minus the inverse sequent of X. And we can see that that's exactly the same as the right hand side of the equation that we started with once we rearranged it. So here we can use substitution, we'll bring down this inverse sine of X equals and on the right where we have pi divided by two minus the inverse C can of X, we'll substitute what that's equal to which is the inverse kant of X. So now we have the inverse sine of X equals the inverse kosic can of X. And we recall from previous lessons that sign and cos can't are inverse or excuse me, not inverse reciprocal functions. So how does that work when they are inverse functions? Here? We'll do a little bit of work with this. If we take the inverse sign of X and set that equal to an angle theta, we recall from lessons on inverse functions that another way to say that is that the sign of angle theta is equal to X. Now we also know that the sign of an angle theta is equal to for its reciprocal function one divided by the cosecant of angle theta. And again, we already said that that's equal to X. So if we look at the right part of that one divided by the K can of theta equals X, if we were to solve for the Kosi can of theta, that would be that the kant of theta equals one divided by X. So if we're going back to an inverse function here, we would know that the inverse cosecant of what's on the right hand side, one divided by X would be equal to angle theta. And so these two both being equal to theta are equal to each other. The inverse sign of X equals the inverse cosecant of one divided by X. Now again, we can use substitution where we've got the inverse sign of X, we can substitute in the inverse cosecant of one divided by X. And that's equal to bringing down the right hand side, the cos seeking or excuse me, the inverse cosecant of X. And now if the inverse coin of something is equal to the inverse coin of something, then we can say that those somes are equal to each other. So here we know that one divided by X equals X and we can solve for X here. What we can do is multiply both sides by X and then we'll have one equals X squared. We can take the square root of both sides and we will have X equals plus or minus one. So here we have our solution that X equals both negative one and positive one. And we look at our answer choices and this matches with answer choice. See, well done. We'll catch you on the next one.

Solve each equation for exact solutions.
cos⁻¹ x + tan⁻¹ x = π/2

Key Concepts

Inverse Trigonometric Functions

Trigonometric Identities

Quadrants and Angle Relationships

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