Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.
sec² θ tan θ = 2 tan θ

Verified step by step guidance

1. Start by writing down the given equation: \(\sec^{2} \theta \tan \theta = 2 \tan \theta\).
2. Recognize that \(\tan \theta\) is a common factor on both sides. To simplify, subtract \(2 \tan \theta\) from both sides to get: \(\sec^{2} \theta \tan \theta - 2 \tan \theta = 0\).
3. Factor out \(\tan \theta\) from the left side: \(\tan \theta (\sec^{2} \theta - 2) = 0\).
4. Set each factor equal to zero to find possible solutions: (1) \(\tan \theta = 0\) and (2) \(\sec^{2} \theta - 2 = 0\).
5. For (2), rewrite \(\sec^{2} \theta\) in terms of \(\tan^{2} \theta\) using the identity \(\sec^{2} \theta = 1 + \tan^{2} \theta\), then solve for \(\tan \theta\). Finally, find all \(\theta\) in \([0^\circ, 360^\circ)\) that satisfy these conditions.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, recognizing that sec²θ = 1 + tan²θ helps simplify and solve the equation by expressing all terms in a single trigonometric function.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within a specified interval. This often requires factoring, using identities, and considering the periodic nature of functions like tangent and secant.

Interval and Solution Set for Angles

When solving trigonometric equations over [0°, 360°), it is essential to find all solutions within one full rotation of the unit circle. Understanding how to interpret and express solutions as exact values or decimal approximations ensures completeness and accuracy.