# Question #c42b1

Jun 12, 2016

We will use the following:

• $\cos \left(\alpha + \beta\right) = \cos \left(\alpha\right) \cos \left(\beta\right) - \sin \left(\alpha\right) \sin \left(\beta\right)$
• $\cos \left(\alpha - \beta\right) = \cos \left(\alpha\right) \cos \left(\beta\right) + \sin \left(\alpha\right) \sin \left(\beta\right)$
• $\cos \left({45}^{\circ}\right) = \sin \left({45}^{\circ}\right) = \frac{\sqrt{2}}{2}$

With those, we have

$\cos \left(\theta + {45}^{\circ}\right) = \cos \left(\theta\right) \cos \left({45}^{\circ}\right) - \sin \left(\theta\right) \sin \left({45}^{\circ}\right)$

$= \frac{\sqrt{2}}{2} \cos \left(\theta\right) - \frac{\sqrt{2}}{2} \sin \left(\theta\right)$

$\cos \left(\theta - {45}^{\circ}\right) = \cos \left(\theta\right) \cos \left({45}^{\circ}\right) + \sin \left(\theta\right) \sin \left({45}^{\circ}\right)$

$= \frac{\sqrt{2}}{2} \cos \left(\theta\right) + \frac{\sqrt{2}}{2} \sin \left(\theta\right)$

$\cos \left(\theta + {45}^{\circ}\right) + \cos \left(\theta - {45}^{\circ}\right)$
$= \frac{\sqrt{2}}{2} \cos \left(\theta\right) - \frac{\sqrt{2}}{2} \sin \left(\theta\right) + \frac{\sqrt{2}}{2} \cos \left(\theta\right) + \frac{\sqrt{2}}{2} \sin \left(\theta\right)$
$= 2 \left(\frac{\sqrt{2}}{2} \cos \left(\theta\right)\right)$
$= \sqrt{2} \cos \left(\theta\right)$