Solve each equation for exact solutions.
tan⁻¹ x = cot⁻¹...

Question

Solve each equation for exact solutions.

tan⁻¹ x = cot⁻¹ 7/5

Accepted Answer

Hey, everyone here we are asked to find the exact solution of the equation. Tangent inverse of X is equal to cotangent inverse of 17 divided by three. Here we have four answer choice options. A X is equal to negative one BX is equal to three divided by 17 CX is equal to three and D no solution. So to find the exact solution of our equation, we need to solve four X. Well, before we can solve four X, we can notice that the right hand side is rather complex. So we can begin by letting the variable U be equal to the right hand side. So U is equal to cotangent inverse of 17 divided by three. And so now we want to rewrite our given equation utilizing this identified variable U. And so rewriting, we have tangent inverse of X is equal to U. And now with this rewritten equation, we can much more easily solve four X by simply taking the tangent of both sides. And so taking the tangent of both sides, we notice on the left hand side, tangent and tangent inverse cancel each other out. So we are just left with X is equal to tangent of U. So now we want to evaluate our current equation since we have already solved four X. And so to find tangent of U, we want to refer back to our identified variable U where we know U is equal to cotangent inverse of 17 divided by three. Well, this equation tells us that if we have cotangent of U, we know that this is equivalent to 17 divided by three. And so now to find tangent of you, we just need to recall that tangent of X is equivalent to one divided by cotangent of X. And so now with this formula and our information that we already have, we can find tangent of you. So we know that tangent of you is equivalent to one divided by cotangent of you. And we have our value for cotangent of you. So we know that this expression is equivalent to one divided by 17, divided by three. And now just simplifying, we have found that tangent of you is equal to three divided by 17. And now just going back to our rewritten equation, we again had X is equal to tangent of you. Well, we have found our value for tangent of you. So just substituting it in, we have X is equal to three divided by 17. And with this, we have found the exact value for our given equation. And so we are left with answer choice B where again X is equal to three divided by 17. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each equation for exact solutions.
tan⁻¹ x = cot⁻¹ 7/5

Key Concepts

Inverse Trigonometric Functions

Trigonometric Identities

Relationship Between Tangent and Cotangent

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