watch the video and then solve the problem given below.\nclick here to watch the video.\nuse a half - angle…

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})