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

Verified step by step guidance

1. Start by recalling the definitions of the inverse secant and inverse cosine functions: sec⁻¹(x) is the angle θ such that sec(θ) = x, and cos⁻¹(y) is the angle φ such that cos(φ) = y.
2. Express sec(θ) in terms of cos(θ): since sec(θ) = 1/cos(θ), we can rewrite the equation sec(θ) = x as 1/cos(θ) = x.
3. Rearrange the equation to find cos(θ): multiply both sides by cos(θ) to get 1 = x * cos(θ), which leads to cos(θ) = 1/x.
4. Now, relate θ to φ: since cos(φ) = 1/x, we can say that φ = cos⁻¹(1/x).
5. Finally, since both θ and φ represent the same angle in terms of their definitions, we conclude that sec⁻¹(x) = cos⁻¹(1/x).