Question

watch the video and then solve the problem given below.
click here to watch the video.
use a half - angle identity to find the exact value.
cos 67.5°
cos 67.5°=
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall the half - angle formula

The half - angle formula for cosine is \(\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}\). Since \(67.5^{\circ}=\frac{135^{\circ}}{2}\) and \(67.5^{\circ}\) is in the first quadrant (\(\cos\theta> 0\) when \(\theta\in(0^{\circ},90^{\circ})\)), we use the positive form of the formula. Here \(\alpha = 135^{\circ}\) and \(\cos135^{\circ}=-\frac{\sqrt{2}}{2}\).

Step2: Substitute into the formula

Substitute \(\alpha = 135^{\circ}\) into \(\cos\frac{\alpha}{2}=\sqrt{\frac{1+\cos\alpha}{2}}\). We get \(\cos67.5^{\circ}=\sqrt{\frac{1+\cos135^{\circ}}{2}}=\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}\).

Step3: Simplify the expression

Simplify \(\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}=\sqrt{\frac{2 - \sqrt{2}}{4}}=\frac{\sqrt{2-\sqrt{2}}}{2}\).

Answer:

\(\frac{\sqrt{2-\sqrt{2}}}{2}\)