Exploring Trigonometric Identities: If AB 45°, What is the Value of 1/(tanA) 1/(tanB)?

Understanding Trigonometric Identities

Trigonometric identities are fundamental relationships between trigonometric functions that can be used to simplify expressions and solve equations. One such identity is the tangent of the sum of two angles, which provides the basis for solving problems like the one presented below. This article will delve into the specific case where AB 45° and explore what the value of 1/(tanA) 1/(tanB) would be. Let's break down the problem step-by-step.

Step-by-Step Solution

Given that AB 45°, we start by applying the tangent to both angles on the left-hand side of the equation.

Given: A B 45° implies tan(A B) (tanA tanB)/(1 - tanA tanB)

Substituting the value of tan45° on the right-hand side, we have:

tan(A B) tan45° 1 implies (tanA tanB)/(1 - tanA tanB) 1

From this, we can derive the following steps:

(tanA tanB) 1 - tanA tanB tanA tanB 1 - tanA tanB 1 1 2 - 1 1 2 1/(tanA) 1/(tanB) 2 Thus, the solution to the problem is 1/(tanA) 1/(tanB) 2.

Applying the Identity to a Different Case

Let's consider another example where AB 45°. We can apply the same steps as before, starting with the identity:

tan(A B) (tanA tanB)/(1 - tanA tanB)

Given that AB π/4 45°, we apply the tangent function on both sides:

tan(45°) 1 implies (tanA tanB)/(1 - tanA tanB) 1

From this, we can derive:

(tanA tanB) 1 - tanA tanB tanA tanB 1 - tanA tanB 1 1 2 - 1 1 1 2 1/(tanA) 1/(tanB) 2 Again, the solution is 1/(tanA) 1/(tanB) 2.

Conclusion

In conclusion, we have explored the problem of finding the value of 1/(tanA) 1/(tanB) given that AB 45°. Through the application of trigonometric identities and the tangent of the sum of two angles, we derived that the value of 1/(tanA) 1/(tanB) is 2. This solution is consistent across various examples, showcasing the power and applicability of trigonometric identities in solving complex problems.