Question

$\tan \left(\pi \cos \theta \right)=\cot \left(\pi \sin \theta \right)$ then the value(s) of $\cos \left(\theta -\frac{\pi }{4}\right)$ is (are)

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

Answer 1

Answer: (C)

$\pm \frac{1}{2\sqrt{2}}$

Given that, $\tan \left(\pi \cos \theta \right)=\cot \left(\pi \sin \theta \right)$

$\frac{\sin \left(\pi \cos \theta \right)}{\cos \left(\pi \cos \theta \right)}=\frac{\cos \left(\pi \sin \theta \right)}{\sin \left(\pi \sin \theta \right)}$

$\sin \left(\pi \cos \theta \right)\sin \left(\pi \sin \theta \right)-\cos \left(\pi \sin \theta \right)\cos \left(\pi \cos \theta \right)=0$

$\cos (\pi \cos \theta +\pi \sin \theta )=0$

$\pi \cos \theta +\pi \sin \theta =\pm \pi /2$

$\cos \theta +\sin \theta =\pm \frac{1}{2}$

$\frac{\cos \theta }{\sqrt{2}}+\frac{\sin \theta }{\sqrt{2}}=\pm \frac{1}{2\sqrt{2}}$

$\cos \theta \cos \frac{\pi }{4}+\sin \theta \sin \frac{\pi }{4}=\pm \frac{1}{2\sqrt{2}}$

$\cos (\theta -\pi /4)=\pm 1/\sqrt{2}$