In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). tan x + sec x = 1

Verified step by step guidance

1. Start with the given equation: \(\tan x + \sec x = 1\).
2. Recall the definitions of tangent and secant in terms of sine and cosine: \(\tan x = \frac{\sin x}{\cos x}\) and \(\sec x = \frac{1}{\cos x}\).
3. Rewrite the equation using these definitions: \(\frac{\sin x}{\cos x} + \frac{1}{\cos x} = 1\).
4. Combine the terms on the left-hand side over a common denominator: \(\frac{\sin x + 1}{\cos x} = 1\).
5. Multiply both sides of the equation by \(\cos x\) (noting \(\cos x \neq 0\)) to get: \(\sin x + 1 = \cos x\). Then, rearrange to isolate terms and use Pythagorean identities to solve for \(x\) in the interval \([0, 2\pi)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, identities like sec x = 1/cos x and tan x = sin x/cos x help transform the equation into a solvable form by expressing all terms in sine and cosine.

Solving Trigonometric Equations

Solving trigonometric equations involves manipulating the equation using identities and algebraic techniques to isolate the variable. After simplification, solutions are found by determining the angles that satisfy the equation within the given interval [0, 2π).

Domain and Interval Considerations

When solving trigonometric equations, it is essential to consider the specified interval, here [0, 2π), to find all valid solutions. Some solutions may be extraneous or outside the interval, so checking each candidate solution against the domain ensures correctness.