Q64 / 160 · AP EAMCET 2013 · Mathematics

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

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

Solution

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

Hence, the correct answer is option: 'A'.

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