If $\tan\left(142\frac{1}{2}°\right) = 2 + \sqrt{2} - \sqrt{\mu} - \sqrt{\lambda}$, then $\mu + \lambda =$
Step-by-Step Solution
Key Concept: Recognize complementary and supplementary angle relationships, then systematically apply tangent subtraction and rationalization.
We simplify $\tan(142.5°) = -\tan(37.5°) = -\tan(60° - 22.5°)$ using angle subtraction. Applying tangent subtraction formulas and rationalizing, we get expressions involving $\sqrt{2}$ and $\sqrt{3}$. After algebraic manipulation with nested radicals, we identify $\mu = 3$ and $\lambda = 6$, yielding $\mu + \lambda = 9$.
Correct Answer: 9