MCQ
$n \in \mathbb{N}$ માટે, જો $\cot ^{-1} 3+\cot ^{-1} 4+\cot ^{-1} 5+\cot ^{-1} n=\frac{\pi}{4}$ હોય, તો $n=$............
- A$70$
- B$56$
- C$10$
- ✓$47$
$ \tan ^{-1} \frac{1}{3}+\tan ^{-1} \frac{1}{4}+\tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{n}=\frac{\pi}{4} $
$ \tan ^{-1}\left(\frac{46}{48}\right)+\tan ^{-1} \frac{1}{n}=\frac{\pi}{4} $
$ \tan ^{-1}\left(\frac{23}{24}\right)+\tan ^{-1} \frac{1}{n}=\frac{\pi}{4} $
$ \tan ^{-1} \frac{1}{n}=\tan ^{-1} 1-\tan ^{-1} \frac{23}{24} $
$ \tan ^{-1} \frac{1}{n}=\tan ^{-1}\left(\frac{1-\frac{23}{24}}{1+\frac{23}{24}}\right) $
$ \tan ^{-1} \frac{1}{n}=\tan ^{-1}\left(\frac{1}{\frac{24}{47}}\right. $
$ \tan ^{-1} \frac{1}{n}=\tan ^{-1} \frac{1}{47} $
$ n=47$
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