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MCQ

If $\sin \left(\frac{\pi}{4} \cot \theta\right)=\cos \left(\frac{\pi}{4} \tan \theta\right)$, then $\theta=$

✓
$n \pi+\frac{\pi}{4}$
B
$2 n \pi \pm \frac{\pi}{4}$
C
$n \pi-\frac{\pi}{4}$
D
$2 n \pi \pm \frac{\pi}{6}$

✓

Answer

Correct option: A.

$n \pi+\frac{\pi}{4}$

(A) $\sin \left(\frac{\pi}{4} \cot \theta\right)=\cos \left(\frac{\pi}{4} \tan \theta\right)$
$\Rightarrow \sin \left(\frac{\pi}{4} \cot \theta\right)=\sin \left(\frac{\pi}{2}-\frac{\pi}{4} \tan \theta\right)$
$\begin{array}{l}\Rightarrow \frac{\pi}{4} \cot \theta=\frac{\pi}{2}-\frac{\pi}{4} \tan \theta \\ \Rightarrow \tan \theta+\cot \theta=2\end{array}$
$\Rightarrow \frac{\sin \theta}{\cos \theta}+\frac{\cos \theta}{\sin \theta}=2$
$\begin{array}{l}\Rightarrow \frac{1}{\sin \theta \cos \theta}=2 \\ \Rightarrow \sin 2 \theta=1\end{array}$
$\Rightarrow 2 \theta=(4 n+1) \frac{\pi}{2} \quad \ldots[$ Using Shortcut $1(i)]$
$\Rightarrow \theta= n \pi+\frac{\pi}{4}$

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