Question

use a half - angle identity to find the exact value of the following expression.
tan 112.5°
which of the following is the correct value of tan 112.5°?
○ a. 1 - √2
○ b. 1 + √2
○ c. - 1 + √2
○ d. - 1 - √2

Explanation:

Step1: Apply the half - angle formula for tangent

The half - angle formula for tangent is \(\tan\frac{\alpha}{2}=\frac{1 - \cos\alpha}{\sin\alpha}\) or \(\tan\frac{\alpha}{2}=-\sqrt{\frac{1-\cos\alpha}{1 + \cos\alpha}}\). Since \(112.5^{\circ}=\frac{225^{\circ}}{2}\) and \(112.5^{\circ}\) is in the second quadrant (\(\tan\theta<0\) in the second quadrant), we use \(\tan\frac{\alpha}{2}=-\sqrt{\frac{1-\cos\alpha}{1+\cos\alpha}}\) with \(\alpha = 225^{\circ}\).
We know that \(\cos225^{\circ}=-\frac{\sqrt{2}}{2}\) and \(\sin225^{\circ}=-\frac{\sqrt{2}}{2}\).
Substitute \(\cos\alpha=-\frac{\sqrt{2}}{2}\) into \(\tan\frac{\alpha}{2}=-\sqrt{\frac{1-\cos\alpha}{1+\cos\alpha}}\):

$$
LATEXBLOCK0
$$

Step2: Rationalize the denominator

Rationalize the denominator of \(\frac{2+\sqrt{2}}{2 - \sqrt{2}}\) by multiplying the numerator and denominator by \(2+\sqrt{2}\):

$$
LATEXBLOCK1
$$

So \(\tan112.5^{\circ}=-\sqrt{3 + 2\sqrt{2}}\).
We can also use another form of the half - angle formula \(\tan\frac{\alpha}{2}=\frac{1-\cos\alpha}{\sin\alpha}\).
Substitute \(\alpha = 225^{\circ}\), \(\cos225^{\circ}=-\frac{\sqrt{2}}{2}\), \(\sin225^{\circ}=-\frac{\sqrt{2}}{2}\) into \(\tan\frac{\alpha}{2}=\frac{1-\cos\alpha}{\sin\alpha}\):

$$
LATEXBLOCK2
$$

Answer:

D. \(-1-\sqrt{2}\)