Solve each equation for exact solutions.
sin⁻¹ x - tan⁻¹ 1 = -π/4

Verified step by step guidance

1. Recognize that \( \sin^{-1} x \) represents the angle whose sine is \( x \), and \( \tan^{-1} 1 \) represents the angle whose tangent is 1.
2. Recall that \( \tan^{-1} 1 = \frac{\pi}{4} \) because the tangent of \( \frac{\pi}{4} \) is 1.
3. Substitute \( \tan^{-1} 1 \) with \( \frac{\pi}{4} \) in the equation: \( \sin^{-1} x - \frac{\pi}{4} = -\frac{\pi}{4} \).
4. Add \( \frac{\pi}{4} \) to both sides of the equation to isolate \( \sin^{-1} x \): \( \sin^{-1} x = 0 \).
5. Determine the value of \( x \) such that \( \sin^{-1} x = 0 \), which means \( x = \sin(0) \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹ (arcsin) and tan⁻¹ (arctan), are used to find angles when given a ratio of sides in a right triangle. For example, sin⁻¹(x) gives the angle whose sine is x, while tan⁻¹(1) gives the angle whose tangent is 1, which is π/4. Understanding these functions is crucial for solving equations involving angles.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identity, sin²(θ) + cos²(θ) = 1, and angle addition formulas. These identities can simplify complex equations and help in finding exact solutions for trigonometric equations.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the values of the variable that satisfy the equation. This often requires isolating the trigonometric function and using inverse functions or identities to find the angle solutions. In the given equation, manipulating the terms and applying the properties of inverse functions will lead to the exact solutions.