Solve each equation for all exact solutions, in degrees.
tan θ - sec θ = 1

Verified step by step guidance

1. Start with the given equation: \(\tan \theta - \sec \theta = 1\).
2. Recall the definitions of tangent and secant in terms of sine and cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\sec \theta = \frac{1}{\cos \theta}\).
3. Rewrite the equation using these definitions: \(\frac{\sin \theta}{\cos \theta} - \frac{1}{\cos \theta} = 1\).
4. Combine the terms on the left side over the common denominator \(\cos \theta\): \(\frac{\sin \theta - 1}{\cos \theta} = 1\).
5. Multiply both sides by \(\cos \theta\) (noting \(\cos \theta \neq 0\)) to get \(\sin \theta - 1 = \cos \theta\), then rearrange to \(\sin \theta - \cos \theta = 1\) and proceed to solve for \(\theta\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Key identities like sec θ = 1/cos θ and tan²θ + 1 = sec²θ help transform and simplify equations, making it easier to solve for θ.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle values that satisfy the equation within a given domain. This often requires algebraic manipulation and using inverse trigonometric functions to find exact solutions.

General Solutions in Degrees

When solving trigonometric equations, the general solution accounts for all possible angles that satisfy the equation, not just principal values. For tangent and secant functions, solutions repeat every 180° or 360°, so adding multiples of these periods ensures all exact solutions are found.