Solving the Equation arctan(x - 1) / arctan(x) / arctan(x 1) π/2

In the realm of trigonometric equations, the equation arctan(x - 1) / arctan(x) / arctan(x 1) π/2 presents a unique challenge and requires careful manipulation and analysis. This article will guide you through the process of solving this complex equation using a systematic approach, demonstrating the importance of algebraic techniques in trigonometry.

Understanding the Problem

When dealing with complex trigonometric equations, it is often helpful to begin by simplifying the notation and setting up the equation in a more manageable form. For the given problem, we introduce the more common notation for the arctangent function, denoted as arctan(x), to cleanse the equation:

arctan(x - 1) - arctan(x) - arctan(x 1) π/2

Transforming the Equation

The function f(x) arctan(x - 1) - arctan(x) - arctan(x 1) is increasing, implying that the equation f(x) π/2 has at most one solution. For simplicity, we can rewrite the equation as:

arctan(x - 1) - arctan(x) π/2 - arctan(x 1)

To solve this equation, we will employ a common trigonometric identity involving the tangent of a sum and difference. Taking the tangent of both sides of the equation gives:

tan(arctan(x - 1) - arctan(x)) tan(π/2 - arctan(x 1))

Using the tangent subtraction formula, we get:

tan(arctan(x - 1) - arctan(x)) (x - 1 - x) / (1 (x - 1)x)

Simplifying the right-hand side, we have:

tan(π/2 - arctan(x 1)) cot(arctan(x 1)) 1 / (x 1)

Therefore, equating the two expressions, we obtain:

(x - 1 - x) / (1 (x - 1)x) 1 / (x 1)

This simplifies to:

-1 / (1 (x - 1)x) 1 / (x 1)

Further simplifying, we get:

-1 / (1 x^2 - x) 1 / (x 1)

Multiplying both sides by the denominators, we obtain:

- (x 1) 1 x^2 - x

Expanding and simplifying, we get:

-x - 1 1 x^2 - x

0 2 x^2 - 1

0 x^2 1

2x^2 2 - x^2

3x^2 2

x^2 2/3

x ±√(2/3)

Validating the Solution

Given that the function arctan(x - 1) - arctan(x) - arctan(x 1) is increasing, the equation can have at most one solution. To validate the obtained solution, we substitute x -√(2/3) into the original equation:

Left-hand side: arctan(-√(2/3) - 1) - arctan(-√(2/3)) - arctan(-√(2/3) 1)

Since the arctangent function is negative for negative inputs, substituting x -√(2/3) would yield a negative value on the left-hand side, which contradicts the right-hand side π/2. Therefore, the negative solution must be discarded.

Conclusion

The unique solution to the equation arctan(x - 1) / arctan(x) / arctan(x 1) π/2 is:

x √(2/3)

Alternative Solutions

While the above solution is obtained through systematic algebraic manipulation, there are other methods to explore. For instance, some advanced mathematical software like Wolfram Alpha can simplify and solve this equation using trigonometric identities. Exploring such software can provide additional insights and validate the derived solution.

References

For further study, refer to the following resources:

Wikipedia: Arctangent AoPS Wiki: Inverse Trigonometric Functions

Understanding such equations not only enhances one's trigonometric skills but also provides a deeper appreciation for the interconnectedness of different mathematical concepts. Happy solving!