Question

if \\( \tan \frac { \pi } { 8 } = \sqrt { a } - b \\), then, by using a half - angle formula, find
\\( a = \\)
\\( b = \\)

Explanation:

Step1: Use the half - angle formula for tangent

The half - angle formula for tangent is \(\tan\frac{\alpha}{2}=\frac{1 - \cos\alpha}{\sin\alpha}\). Let \(\alpha=\frac{\pi}{4}\), then \(\frac{\alpha}{2}=\frac{\pi}{8}\).
We know that \(\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\) and \(\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}\).
Substitute into the formula: \(\tan\frac{\pi}{8}=\frac{1-\cos\frac{\pi}{4}}{\sin\frac{\pi}{4}}\).

Step2: Simplify the expression

$$
LATEXBLOCK0
$$

Answer:

\(A = 2\), \(B = 1\)