Question

use a half - angle formula to find the exact value of the following expression.
(\tan 157.5^{circ})
(\tan 157.5^{circ}=square)
(simplify your answer, including any radicals. use integers or fractions for any

Explanation:

Step1: Determine the half - angle formula

The half - angle formula for tangent is \(\tan\frac{\alpha}{2}=\frac{1 - \cos\alpha}{\sin\alpha}\) (when \(\frac{\alpha}{2}\) is in the second quadrant, \(\tan\frac{\alpha}{2}<0\)). Here \(\frac{\alpha}{2}=157.5^{\circ}\), so \(\alpha = 315^{\circ}\).

Step2: Substitute \(\alpha = 315^{\circ}\) into the formula

We know that \(\cos315^{\circ}=\frac{\sqrt{2}}{2}\) and \(\sin315^{\circ}=-\frac{\sqrt{2}}{2}\).

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Answer:

\(1-\sqrt{2}\)