The value of ( _ r = 1 ^ ^ - 1 ( 4 4 r^2 + 3 ) ) is equal to - Vidyadip

MCQ

The value of $\cot \left( {\sum\limits_{r = 1}^\infty {{{\tan }^{ - 1}}\left( {\frac{4}{{4{r^2} + 3}}} \right)} } \right)$ is equal to

A
$1$
✓
$\frac{1}{2}$
C
$2$
D
$\frac{1}{4}$

✓

Answer

Correct option: B.

$\frac{1}{2}$

b
$\tan ^{-1}\left(\frac{4}{4 r^{2}+3}\right)=\tan ^{-1}\left(r+\frac{1}{2}\right)-\tan ^{-1}\left(r-\frac{1}{2}\right)$

$\sum\limits_{r = 1}^n {{{\tan }^{ - 1}}\left( {\frac{4}{{4{r^2} + 3}}} \right)} = {\cot ^{ - 1}}\left( {\frac{1}{2}} \right)$

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