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True-or-False :If $\tan(\pi \cos \theta)=\cot (\pi \sin \theta)$,then $\cos(\theta-\large\frac{\pi}{4})$$=\pm \large\frac{1}{2\sqrt 2}$

$\begin{array}{1 1}(A)\;\text{True}\\(B)\;\text{False}\end{array} $

1 Answer

$\tan (\pi\cos \theta)=\cot (\pi\sin \theta)$
$\tan (\pi\cos \theta)=-\tan (\large\frac{\pi}{2}-$$\pi\sin \theta)$
$\large\frac{\tan (\pi\cos \theta)}{\tan (\Large\frac{\pi}{2}-\normalsize \pi\sin \theta)}$$=-1$
If $\tan (\pi\cos \theta)=1$ then
$\tan (\large\frac{\pi}{2}$$-\pi\sin \theta)=-1=\tan \large\frac{3\pi}{4}$
$\therefore \pi\cos \theta=\large\frac{\pi}{4}$
$\cos \theta=\large\frac{1}{4}$
$\large\frac{\pi}{2}$$-\pi \sin \theta=\large\frac{3\pi}{4}$
$-\pi \sin \theta=\large\frac{3\pi}{4}-\frac{\pi}{2}$
$-\pi \sin \theta=\large\frac{3\pi-2\pi}{4}=\frac{2\pi}{4}$
$\sin \theta=-\large\frac{1}{4}$
$\tan (\pi\cos \theta)=-1=\tan \large\frac{3\pi}{4}$
$\tan (\large\frac{\pi}{2}$$-\pi\sin\theta)=1=\tan \large\frac{\pi}{4}$
$\pi\cos \theta=\large\frac{3\pi}{4}$
$\cos \theta=\large\frac{3}{4}$
Hence the given statement is true
answered May 20, 2014 by sreemathi.v
 

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