Solve each equation for exact solutions.
cos⁻¹ x + tan⁻¹ x = π/2

Verified step by step guidance

1. Recognize that the equation is given as \(\cos^{-1} x + \tan^{-1} x = \frac{\pi}{2}\). Our goal is to find all values of \(x\) that satisfy this equation exactly.
2. Recall the identity involving inverse trigonometric functions: if \(\cos^{-1} x + \sin^{-1} x = \frac{\pi}{2}\), then we can try to relate \(\tan^{-1} x\) to \(\sin^{-1} x\) or \(\cos^{-1} x\) to simplify the expression.
3. Use the substitution \(\theta = \cos^{-1} x\), which implies \(x = \cos \theta\) and \(\theta \in [0, \pi]\). Then rewrite the equation as \(\theta + \tan^{-1}(\cos \theta) = \frac{\pi}{2}\).
4. Isolate \(\tan^{-1}(\cos \theta)\) to get \(\tan^{-1}(\cos \theta) = \frac{\pi}{2} - \theta\). Then take the tangent of both sides to obtain \(\cos \theta = \tan\left(\frac{\pi}{2} - \theta\right)\).
5. Use the co-function identity \(\tan\left(\frac{\pi}{2} - \theta\right) = \cot \theta = \frac{\cos \theta}{\sin \theta}\). Substitute this back to get \(\cos \theta = \frac{\cos \theta}{\sin \theta}\). From here, solve for \(\theta\) and then find \(x = \cos \theta\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as cos⁻¹(x) and tan⁻¹(x), return the angle whose trigonometric ratio equals x. They are used to find angles from known ratios and have specific ranges to ensure they are functions. Understanding their definitions and ranges is essential for solving equations involving them.

Trigonometric Identities Involving Inverse Functions

Certain identities relate inverse trigonometric functions, such as cos⁻¹(x) + sin⁻¹(x) = π/2. Recognizing or deriving similar identities helps simplify and solve equations involving sums of inverse trig functions. These identities often rely on complementary angle relationships.

Solving Equations with Inverse Trigonometric Expressions

Solving equations like cos⁻¹(x) + tan⁻¹(x) = π/2 requires manipulating inverse trig expressions, possibly by expressing one function in terms of another or using substitution. Understanding domain restrictions and exact values of inverse functions aids in finding precise solutions.