Use the Half-angle Formulas to find the exat value of each expression.

Use the Half-angle Formulas to find the exat value of each expression.
$$\displaystyle{{\tan{{22.5}}}^{\circ}}$$

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Margot Mill
Trigonometry:
$$\displaystyle{\tan{{\left({22.5}^{\circ}\right)}}}={\tan{{t}}}$$
$$\displaystyle{\tan{{2}}}{t}={{\tan{{45}}}^{\circ}=}{1}$$
Use trig identify:
$$\displaystyle{\tan{{\left({2}{t}\right)}}}={\frac{{{2}{\tan{{t}}}}}{{{1}-{{\tan}^{{2}}{t}}}}}$$
$$\displaystyle{\frac{{{2}{\tan{{t}}}}}{{{1}-{{\tan}^{{2}}{t}}}}}={1}$$
$$\displaystyle{{\tan}^{{2}}{t}}+{2}{\tan{{t}}}-{1}={0}$$
$$\displaystyle{\tan{{t}}}={\frac{{-{2}\pm\sqrt{{{4}+{4}}}}}{{{2}}}}$$
$$\displaystyle={\frac{{-{2}\pm{2}\sqrt{{{2}}}}}{{{2}}}}$$
$$\displaystyle=-{1}\pm\sqrt{{{2}}}$$
Since $$\displaystyle{\tan{{22.5}}}$$ is positive, then take the positive answer:
$$\displaystyle{\tan{{\left({22.5}\right)}}}=-{1}+\sqrt{{{2}}}$$