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Which one of the following is not correct?
Options:
Solution:
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Verified Answer
The correct answer is:
$\cos e c^{-1}\{\operatorname{cosec}(7 \pi / 4)\}=\pi / 4$
Option (a) $\sin ^{-1}\left(\sin \frac{5 \pi}{4}\right)=\frac{-\pi}{4}$
$\Rightarrow \sin \frac{5 \pi}{4}=\sin \left(\frac{-\pi}{4}\right)$
$\Rightarrow \sin \left(\pi+\frac{\pi}{4}\right)=-\sin \frac{\pi}{4}$
$\Rightarrow-\sin \left(\frac{\pi}{4}\right)=-\sin \left(\frac{\pi}{4}\right)$
Hence it is correct Option (b)
$\sec ^{-1}\left(\sec \frac{5 \pi}{4}\right)=\frac{3 \pi}{4}$
$\Rightarrow \sec \frac{5 \pi}{4}=\sec \frac{3 \pi}{4}$
$\Rightarrow \sec \left(2 \pi-\frac{3 \pi}{4}\right)=\sec \frac{3 \pi}{4}$
$\Rightarrow \sec \frac{3 \pi}{4}=\sec \frac{3 \pi}{4}$
Hence, it is correct
Option (c)
$\tan ^{-1}\left(\tan \frac{5 \pi}{4}\right)=\frac{\pi}{4}$
$\Rightarrow \tan \frac{5 \pi}{4}=\tan \frac{\pi}{4}$
$\Rightarrow \tan \left(\pi+\frac{\pi}{4}\right)=\tan \frac{\pi}{4}$
$\Rightarrow \tan \frac{\pi}{4}=\tan \frac{\pi}{4}$
Hence it is correct Option (d). $\operatorname{cosec}^{-1}\left(\operatorname{cosec} \frac{7 \pi}{4}\right)=\frac{\pi}{4}$
$\Rightarrow \operatorname{cosec} \frac{7 \pi}{4}=\operatorname{cosec} \frac{\pi}{4}$
$\Rightarrow \operatorname{cosec}\left(2 \pi-\frac{\pi}{4}\right)=\operatorname{cosec} \frac{\pi}{4}$
$\Rightarrow-\operatorname{cosec} \frac{\pi}{4}=\operatorname{cosec} \frac{\pi}{4}$
Hence it is not correct.
$\Rightarrow \sin \frac{5 \pi}{4}=\sin \left(\frac{-\pi}{4}\right)$
$\Rightarrow \sin \left(\pi+\frac{\pi}{4}\right)=-\sin \frac{\pi}{4}$
$\Rightarrow-\sin \left(\frac{\pi}{4}\right)=-\sin \left(\frac{\pi}{4}\right)$
Hence it is correct Option (b)
$\sec ^{-1}\left(\sec \frac{5 \pi}{4}\right)=\frac{3 \pi}{4}$
$\Rightarrow \sec \frac{5 \pi}{4}=\sec \frac{3 \pi}{4}$
$\Rightarrow \sec \left(2 \pi-\frac{3 \pi}{4}\right)=\sec \frac{3 \pi}{4}$
$\Rightarrow \sec \frac{3 \pi}{4}=\sec \frac{3 \pi}{4}$
Hence, it is correct
Option (c)
$\tan ^{-1}\left(\tan \frac{5 \pi}{4}\right)=\frac{\pi}{4}$
$\Rightarrow \tan \frac{5 \pi}{4}=\tan \frac{\pi}{4}$
$\Rightarrow \tan \left(\pi+\frac{\pi}{4}\right)=\tan \frac{\pi}{4}$
$\Rightarrow \tan \frac{\pi}{4}=\tan \frac{\pi}{4}$
Hence it is correct Option (d). $\operatorname{cosec}^{-1}\left(\operatorname{cosec} \frac{7 \pi}{4}\right)=\frac{\pi}{4}$
$\Rightarrow \operatorname{cosec} \frac{7 \pi}{4}=\operatorname{cosec} \frac{\pi}{4}$
$\Rightarrow \operatorname{cosec}\left(2 \pi-\frac{\pi}{4}\right)=\operatorname{cosec} \frac{\pi}{4}$
$\Rightarrow-\operatorname{cosec} \frac{\pi}{4}=\operatorname{cosec} \frac{\pi}{4}$
Hence it is not correct.
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