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

Verified step by step guidance

1. Recall the relationship between inverse tangent and inverse cotangent functions: \(\tan^{-1} x = \cot^{-1} y\) implies \(x = \frac{1}{y}\), because \(\cot \theta = \frac{1}{\tan \theta}\).
2. Given the equation \(\tan^{-1} x = \cot^{-1} \frac{7}{5}\), rewrite the right side using the reciprocal relationship: \(\cot^{-1} \frac{7}{5} = \tan^{-1} \frac{5}{7}\).
3. Set the expressions equal: \(\tan^{-1} x = \tan^{-1} \frac{5}{7}\), which means \(x = \frac{5}{7}\), considering the principal values of inverse tangent.
4. Remember that inverse tangent and cotangent functions have ranges that might allow for additional solutions differing by \(\pi\), so consider the general solution for \(\tan^{-1} x = \cot^{-1} \frac{7}{5}\) by using the identity \(\cot^{-1} y = \tan^{-1} \frac{1}{y} + k\pi\) for integers \(k\).
5. Write the general solution as \(x = \tan \left( \cot^{-1} \frac{7}{5} + k\pi \right)\) and simplify to find all exact solutions, where \(k\) is any integer.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as tan⁻¹ (arctan) and cot⁻¹ (arccot), return the angle whose trigonometric ratio equals a given value. Understanding their ranges and how to interpret their outputs is essential for solving equations involving these functions.

Relationship Between Tangent and Cotangent

Tangent and cotangent are reciprocal functions, meaning tan(θ) = 1/cot(θ). This relationship helps convert between tan⁻¹ and cot⁻¹ expressions, allowing the equation to be rewritten in a single trigonometric function for easier solution.

Exact Solutions and Principal Values

Exact solutions involve finding precise angle values, often expressed in terms of inverse trig functions or special angles. Since inverse trig functions have principal value ranges, recognizing these ranges ensures the correct angle is identified as the solution.