Question

If $\tan (\pi \cos \theta )=\cot (\pi \sin \theta )$, then a value of $\cos \left(\theta -\frac{\pi }{4}\right)$ among the following is

• A. $\frac{1}{2\sqrt{2}}$
• B. $\frac{1}{\sqrt{2}}$
• C. $\frac{1}{2}$
• D. $\frac{1}{4}$

Answer 1

Answer: (A)

Given, $\tan (\pi \cos \theta )=\cot (\pi \sin \theta )$
$\Rightarrow \tan (\pi \cos \theta )=\tan \left\{\frac{\pi }{2}-\pi \sin \theta \right\}$
$\Rightarrow \pi \cos \theta =\frac{\pi }{2}-\pi \sin \theta$
$\Rightarrow \sin \theta +\cos \theta =\frac{1}{2}$
$\Rightarrow \frac{1}{\sqrt{2}}\sin \theta +\frac{1}{\sqrt{2}}\cos \theta =\frac{1}{2\sqrt{2}}$
$\Rightarrow \cos \theta \cdot \cos \frac{\pi }{4}+\sin \theta \cdot \sin \frac{\pi }{4}=\frac{1}{2\sqrt{2}}$
$\Rightarrow \cos \left(\theta -\frac{\pi }{4}\right)=\frac{1}{2\sqrt{2}}$