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. Rewrite the equation using trigonometric identities: $\tan x=\frac{\sin x}{\cos x}$ and $\sec x=\frac{1}{\cos x}$.
2. Combine the terms over a common denominator: $\frac{\sin x+1}{\cos x}=1$.
3. Multiply both sides by $\cos x$ to eliminate the fraction: $\sin x+1=\cos x$.
4. Rearrange the equation to isolate $\sin x$: $\sin x=\cos x-1$.
5. Use the Pythagorean identity ${\sin }^{2}x+{\cos }^{2}x=1$ to express $\sin x$ in terms of $\cos x$ and solve for $x$ in the interval $[0,2\pi )$.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variable where both sides of the equation are defined. Key identities include the Pythagorean identities, reciprocal identities, and co-function identities. Understanding these identities is crucial for simplifying trigonometric expressions and solving equations, as they allow for the transformation of one function into another.

Tangent and Secant Functions

The tangent function, defined as the ratio of the sine and cosine functions (tan x = sin x / cos x), represents the slope of the angle in a right triangle. The secant function, which is the reciprocal of the cosine function (sec x = 1 / cos x), is used to relate angles to their corresponding sides in a triangle. Recognizing the relationship between these functions is essential for manipulating and solving equations involving them.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the angles that satisfy the equation within a specified interval. This process often requires the use of identities to rewrite the equation in a more manageable form. Techniques such as factoring, using inverse functions, and applying known values of trigonometric functions are commonly employed to isolate the variable and determine the solutions.