Question

The principal value of ${\tan }^{-1}\left(\cot \frac{43\pi }{4}\right)$ is

• A. $-\frac{3\pi }{4}$
• B. $\frac{3\pi }{4}$
• C. $-\frac{\pi }{4}$
• D. $\frac{\pi }{4}$

Answer 1

The correct option is C $-\frac{\pi }{4}$

${\tan }^{-1}\left(\cot \left(43\frac{\pi }{4}\right)\right)$

$={\tan }^{-1}\left(\cot (10\pi +3\frac{\pi }{4})\right)$

$={\tan }^4\left(\cot \left(\frac{3\pi }{-1}\right)\right)$

$={\tan }^4\left(\cot (\frac{\pi }{2}+\frac{\pi }{-1})\right)$

$={\tan }^{-1}(-\tan \frac{\pi }{4})$

$={\tan }^{-1}\left(\tan \left(\frac{-\pi }{4}\right)\right)$

$=\frac{-\pi }{4}$