Final answer:
The exact value of the given expression is √3 / (4 * (1 - √3)).
Step-by-step explanation:
To find the exact value of the expression tan(106) / [1 − tan(106°) + tan(-61°)] / (1 - tan(106°) tan(-61°)), we need to evaluate the trigonometric functions for the given angles. Let's break it down step by step.
Step 1: Evaluate the trigonometric functions for the angles:
- tan(106°) = √3
- tan(-61°) = -√3
Step 2: Substitute the values into the expression:
tan(106) / [1 − tan(106°) + tan(-61°)] / (1 - tan(106°) tan(-61°)) = √3 / [1 - √3 + (-√3)] / [1 - √3 * (-√3)]
Step 3: Simplify the expression:
√3 / [1 - √3 + (-√3)] / [1 - √3 * (-√3)] = √3 / (1 - √3) / (1 + 3) = √3 / (1 - √3) / 4
So, the exact value of the given expression is √3 / (4 * (1 - √3)).