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

Verified step by step guidance

1. Recall the definitions of tangent and cotangent: \(\tan x = \frac{\sin x}{\cos x}\) and \(\cot x = \frac{\cos x}{\sin x}\).
2. Set the equation \(\tan x = \cot x\) and rewrite it using the definitions: \(\frac{\sin x}{\cos x} = \frac{\cos x}{\sin x}\).
3. Cross-multiply to eliminate the fractions: \(\sin^2 x = \cos^2 x\).
4. Use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) to express one function in terms of the other, or recognize that \(\sin^2 x = \cos^2 x\) implies \(\sin^2 x - \cos^2 x = 0\).
5. Rewrite the equation as \(\sin^2 x - \cos^2 x = 0\), which can be factored or recognized as \(\cos 2x = 0\). Solve \(\cos 2x = 0\) over the interval \([0, 2\pi)\) to find the values of \(x\).

Key Concepts

Relationship Between Tangent and Cotangent

Tangent and cotangent are reciprocal trigonometric functions, where tan(x) = sin(x)/cos(x) and cot(x) = cos(x)/sin(x). Understanding their relationship helps in transforming or equating expressions involving these functions.

Solving Trigonometric Equations

Solving equations like tan(x) = cot(x) involves manipulating the equation to find values of x that satisfy it within a given interval, often by using identities or rewriting functions in terms of sine and cosine.

Interval and General Solutions in Trigonometry

Trigonometric functions are periodic, so solutions repeat every 2π. When solving over [0, 2π), it is important to find all unique solutions within this interval, considering the periodicity and domain restrictions.