Evaluate cos(tan^(-1)(1)).

Evaluate cos(tan^(-1)(1)).

Correct Answer: 1/√2

Step 1: Understand that tan^(-1)(1) means we are looking for an angle θ where the tangent of θ equals 1.
Step 2: Recall that the tangent of 45 degrees (or π/4 radians) is 1. So, θ = 45 degrees (or θ = π/4).
Step 3: We need to find cos(θ). We can use the formula for cosine in terms of tangent: cos(θ) = 1 / √(1 + tan^2(θ)).
Step 4: Since we know tan(θ) = 1, we can substitute this into the formula: cos(θ) = 1 / √(1 + 1^2).
Step 5: Simplify the expression: cos(θ) = 1 / √(1 + 1) = 1 / √2.
Step 6: Therefore, the final answer is cos(tan^(-1)(1)) = 1 / √2.

Inverse Trigonometric Functions – Understanding how to evaluate inverse trigonometric functions, specifically tan^(-1) and its relationship to right triangles.
Trigonometric Identities – Applying the identity cos(θ) = 1/√(1 + tan^2(θ)) to find the cosine of an angle derived from the tangent function.