In Exercises 97–116, use the most appropriate method to solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. tan x sec x = 2 tan x

Verified step by step guidance

1. Start with the given equation: \(\tan x \sec x = 2 \tan x\).
2. Bring all terms to one side to set the equation to zero: \(\tan x \sec x - 2 \tan x = 0\).
3. Factor out the common factor \(\tan x\): \(\tan x (\sec x - 2) = 0\).
4. Set each factor equal to zero and solve separately: 1) \(\tan x = 0\) 2) \(\sec x - 2 = 0\).
5. For \(\tan x = 0\), find all \(x\) in \([0, 2\pi)\) where tangent is zero. For \(\sec x - 2 = 0\), rewrite as \(\sec x = 2\), then use the identity \(\sec x = \frac{1}{\cos x}\) to find \(\cos x = \frac{1}{2}\) and solve for \(x\) in \([0, 2\pi)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, recognizing identities like sec x = 1/cos x and the relationship between tan x and sin x/cos x helps simplify and solve the equation efficiently.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all solutions within a given interval. This often requires factoring, using identities, and considering the domain restrictions to find exact or approximate values of x.

Interval and Solution Restrictions

When solving trigonometric equations on a specific interval like [0, 2π), it is essential to find all solutions within that range. Additionally, one must consider where functions are undefined (e.g., sec x undefined when cos x = 0) to exclude invalid solutions.