Verify the identity sec⁻¹ x = cos⁻¹ (1/x), for x ≠ 0.

Verified step by step guidance

1. Start by understanding the definitions: sec⁻¹(x) is the inverse secant function, which gives the angle whose secant is x. Similarly, cos⁻¹(y) is the inverse cosine function, which gives the angle whose cosine is y.
2. Recall the relationship between secant and cosine: sec(θ) = 1/cos(θ). Therefore, if θ = sec⁻¹(x), then sec(θ) = x, which implies cos(θ) = 1/x.
3. Express the angle θ in terms of cosine: Since cos(θ) = 1/x, we can write θ = cos⁻¹(1/x).
4. Thus, if θ = sec⁻¹(x), then θ = cos⁻¹(1/x), which verifies the identity sec⁻¹(x) = cos⁻¹(1/x).
5. Ensure the domain is correct: The identity holds for x ≠ 0, as secant and cosine are defined for these values, and 1/x is valid when x is not zero.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sec⁻¹ (arcsec) and cos⁻¹ (arccos), are used to find angles when given a ratio. The arcsec function returns the angle whose secant is x, while the arccos function returns the angle whose cosine is 1/x. Understanding these functions is crucial for verifying identities involving trigonometric ratios.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. The identity sec(θ) = 1/cos(θ) is fundamental in relating secant and cosine functions. Recognizing and applying these identities is essential for proving relationships between different trigonometric functions.

Domain and Range of Functions

The domain and range of functions define the set of input values (domain) and the set of possible output values (range). For sec⁻¹ x, the domain is x ≤ -1 or x ≥ 1, while for cos⁻¹ (1/x), the domain is restricted to x ≤ -1 or x ≥ 1 as well. Understanding these constraints is vital for ensuring the validity of the identity being verified.