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  4. If sin((π/4) cot θ)= cos((π/4) tan θ),then θ=

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

Trigonometric Functions

Solution:

$\sin\left(\frac{\pi}{4}\,\cot\,\theta\right)=\cos\,\left(\frac{\pi}{4}\,\tan\,\theta\,\right)$
$\Rightarrow \,\sin \left(\frac{\pi}{4}\,\tan\,\theta\right)=\cos\,\left(\frac{\pi}{2}-\frac{\pi}{4}\cot\,\theta\,\right)$
$\Rightarrow \,\frac{\pi}{4}\,\tan\,\theta=\frac{\pi}{2}-\frac{\pi}{4}\,\cot\,\theta$
$\Rightarrow \,\frac{\pi}{4}(\tan\,\theta+\cot\,\theta)=\frac{\pi}{2}$
$\Rightarrow \,\frac{\sin\,\theta}{\cos\,\theta}+\frac{\cos\,\theta}{\sin\,\theta}=2$
$\Rightarrow \,\sin^2\,\theta+\cos^2\,\theta-2\,\sin\,\theta\,\cos\,\theta=0$
$\Rightarrow \,(\sin\,\theta-\cos\,\theta)^2=0\,\Rightarrow \,\sin\,\theta-\cos\,\theta=0$
$\Rightarrow \,\tan\,\theta=1=\frac{\pi}{4} \Rightarrow \,\theta=n\pi\,\pm\,\frac{\pi}{4}\,$ where $n\,\in\,I$