Question

The value of ${\tan }^{-1}\left(\tan 1\right)+{\tan }^{-1}\left(\tan 2\right)+{\tan }^{-1}\left(\tan 3\right)+{\tan }^{-1}\left(\tan 4\right)$ is

• A. $10-3\pi$
• B. $10$
• C. $10+3\pi$
• D. $0$

Answer 1

The correct option is A $10-3\pi$

${\tan }^{-1}\tan 1=1$
${\tan }^{-1}\tan 2=2-\pi$
${\tan }^{-1}\tan 3=3-\pi$
${\tan }^{-1}\tan 4=4-\pi$
So, ${\tan }^{-1}\tan 1+{\tan }^{-1}\tan 2+{\tan }^{-1}\tan 3+{\tan }^{-1}\tan 4=1+2-\pi +3-\pi +4-\pi =10-3\pi$