An inverse tangent identity

b. Prove that tan...

Question

An inverse tangent identity

b. Prove that tan⁻¹ x + tan⁻¹ x(1/x) = π/2, for x > 0.

Accepted Answer

Hello. In this video, we are going to be finding the value of A for the given equation. We are told to find the value of the given inverse trigonometric function when A is greater than 0. The function given to us is tangent inverse of A plus tangent inverse of one divided by A. Now, how can we start to solve for the value of A for the trigonometric function? Well, in order to approach this problem, we are going to use the identity that is the sum of two arctens functions. That identity is going to be tangent inverse of X, plus tangent inverse of Y. And that is going to equal to the value, tangent inverse. Of X + Y divided by 1 minus X Y and this is where X Y is less than 1. So, in order to use this identity, we are going to allow the value of X to be A and the value of Y to be 1 divided by A. By using these values, we can rewrite the statement to be tangent inverse. Of a Plus tangent inverse of one divided by a and by the identity this will equal tangent inverse of a plus 1 divided by a all divided by 1 minus a multiplied by 1 divided by a. So, in order to find the value of A that makes the function true, we are going to simplify the right hand side. So, if we take a look at the denominator, A multiplied by 1 divided by A will give us the value of 1. That will allow us to rewrite the tangent inverse, as tangent inverse of A plus 1 divided by A is divided by 1 minus 1. But there is a small problem here that will give us tangent inverse of an undefined value, because we will get division by zero. Now, if we take tangent inverse of an undefined value, and let's suppose that this equals to the output of A. If we want to solve for the value of A, we can first take the tangent function of both sides of the equation, and rewrite this as tangent of A equal to an undefined value. Now we need to ask ourselves, what is a value that we can plug into the tangent function that will give us an undefined value? And again, the problem tells us that the value of A must be strictly greater than 0. So we need to think about a positive value A to plug into tangent to the regular tangent function. That will give us an undefined value. Well, that is going to be the restriction of the tangent inverse. We know that tangent of pi divided by 2 will produce an undefined value for the tangent function. So what this means is that tangent inverse of a plus tangent inverse of 1 divided by A will be undefined when A is equal to pi divided by 2, and that is going to be the solution to the problem. And with that being said, the overall solution is going to be A. So I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

An inverse tangent identity

b. Prove that tan⁻¹ x + tan⁻¹ x(1/x) = π/2, for x > 0.

Key Concepts

Inverse Tangent Function

Trigonometric Identities

Complementary Angles

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