For $n\in N$, if $\cot^{-1}3+\cot^{-1}4+\cot^{-1}5+\cot^{-1}n=\frac{\pi}{4}$, then $n$ is equal to ____.

JEE Main · 2024 · 06 Apr 2024 · Shift 1 · Inverse Trigonometric Functions

For

n \in N

, if

cot^{- 1} 3 + cot^{- 1} 4 + cot^{- 1} 5 + cot^{- 1} n = \frac{π}{4}

, then

n

is equal to ____.

Answer: 47

Write as

tan^{- 1} \frac{1}{3} + tan^{- 1} \frac{1}{4} + tan^{- 1} \frac{1}{5} + tan^{- 1} \frac{1}{n} = \frac{π}{4}

tan^{- 1} \frac{1}{3} + tan^{- 1} \frac{1}{4} = tan^{- 1} \frac{7}{11}

; adding

tan^{- 1} \frac{1}{5}

gives

tan^{- 1} \frac{23}{24}

.
Then

tan^{- 1} \frac{1}{n} = \frac{π}{4} - tan^{- 1} \frac{23}{24} = tan^{- 1} \frac{1 - 23/24}{1 + 23/24} = tan^{- 1} \frac{1}{47}

, so

n = 47.

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