Question

Find the value of $\cos$ $\theta -\cos 3\theta +\cos 4\theta ,$ when $\theta ={45}^0$

• A. $\frac{\sqrt{3}}{2}$
• B. $\sqrt{3}-1$
• C. $\frac{1}{2}$
• D. $\sqrt{2}-1$

Answer 1

The correct option is D $\sqrt{2}-1$

The value of $\cos \left(\frac{\pi }{4}\right)-\cos \left(\frac{3\pi }{4}\right)+\cos \left(\frac{4\pi }{4}\right)$ is
$\cos \left(\frac{\pi }{4}\right)-\cos (\pi -\frac{\pi }{4})+\cos \left(\pi \right)$
$=\cos \left(\frac{\pi }{4}\right)+\cos \left(\frac{\pi }{4}\right)+\cos \left(\pi \right)$
$=2\cos \left(\frac{\pi }{4}\right)+\cos \left(\pi \right)$
$=\frac{2}{\sqrt{2}}-1$
$=\sqrt{2}-1$
Hence answer is D