Solve each equation over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate.
tan² 2x -1 = 0

Verified step by step guidance

1. Start with the given equation: \(\tan^{2} 2x - 1 = 0\).
2. Rewrite the equation to isolate the squared tangent term: \(\tan^{2} 2x = 1\).
3. Take the square root of both sides, remembering to consider both positive and negative roots: \(\tan 2x = \pm 1\).
4. Solve for \$2x\( by finding all angles where \(\tan \theta = 1\) and \(\tan \theta = -1\) within one full period of tangent, which is \(\pi\). So, \(2x = \frac{\pi}{4} + k\pi\) and \(2x = \frac{3\pi}{4} + k\pi\), where \)k$ is any integer.
5. Divide all solutions by 2 to solve for \(x\), then find all values of \(x\) within the interval \([0, 2\pi)\) by choosing appropriate integer values of \(k\).

Key Concepts

Solving Trigonometric Equations

Solving trigonometric equations involves finding all angle values within a specified interval that satisfy the given equation. This often requires algebraic manipulation and applying inverse trigonometric functions, while considering the periodic nature of trig functions to find all valid solutions.

Properties of the Tangent Function

The tangent function, tan(θ), is periodic with period π and is defined as sin(θ)/cos(θ). Understanding its behavior, including where it is positive or negative and its undefined points, is essential for solving equations involving tan²(θ) and for determining all solutions within a given interval.

Using Double-Angle Identities

Double-angle identities express trigonometric functions of 2x in terms of x, such as tan(2x) = 2tan(x)/(1 - tan²(x)). Recognizing and applying these identities helps simplify equations involving multiple angles, enabling easier solution of equations like tan²(2x) - 1 = 0.