Question:

The value of $\cot\left(\sum\limits^{19}_{n=1} \cot^{-1} \left(1+ \sum\limits^{n}_{p=1} 2p\right)\right) $ is :

Updated On: Aug 15, 2024

$\frac{22}{23}$
$\frac{23}{22}$
$\frac{21}{19}$
$\frac{19}{21}$

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The Correct Option is C

Solution and Explanation

$\cot\left(\sum^{19}_{n=1} \cot^{-1} \left(1+n\left(n+1\right)\right)\right) $
$ \cot\left(\sum^{19}_{n=1} \cot^{-1}\left(n^{2}+n+1\right)\right) = \cot\left(\sum^{19}_{n=1} \tan^{-1} \frac{1}{1+n\left(n+1\right)}\right)$
$ \sum^{19}_{n=1} \left(\tan^{-1}\left(n+1\right)-\tan^{-1}n\right)$
$ \cot\left(\tan^{-1}20 -\tan^{-1}1\right)= \frac{\cot A \cot\beta+1}{\cot\beta-\cot A} $
(Where $\tan A=20, \tan B=1 $ )
$ \frac{1\left(\frac{1}{20}\right)+1}{1- \frac{1}{20}} = \frac{21}{19} $

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Concepts Used:

Properties of Inverse Trigonometric Functions

The elementary properties of inverse trigonometric functions will help to solve problems. Here are a few important properties related to inverse trigonometric functions:

Property Set 1:

Sin⁻¹(x) = cosec⁻¹(1/x), x∈ [−1,1]−{0}
Cos⁻¹(x) = sec⁻¹(1/x), x ∈ [−1,1]−{0}
Tan⁻¹(x) = cot⁻¹(1/x), if x > 0 (or) cot⁻¹(1/x) −π, if x < 0
Cot⁻¹(x) = tan⁻¹(1/x), if x > 0 (or) tan⁻¹(1/x) + π, if x < 0

Property Set 2:

Sin⁻¹(−x) = −Sin⁻¹(x)
Tan⁻¹(−x) = −Tan⁻¹(x)
Cos⁻¹(−x) = π − Cos⁻¹(x)
Cosec⁻¹(−x) = − Cosec⁻¹(x)
Sec⁻¹(−x) = π − Sec⁻¹(x)
Cot⁻¹(−x) = π − Cot⁻¹(x)

Property Set 3:

Sin⁻¹(1/x) = cosec⁻¹x, x≥1 or x≤−1
Cos⁻¹(1/x) = sec⁻¹x, x≥1 or x≤−1
Tan⁻¹(1/x) = −π + cot⁻¹(x)

Property Set 4:

Sin⁻¹(cos θ) = π/2 − θ, if θ∈[0,π]
Cos⁻¹(sin θ) = π/2 − θ, if θ∈[−π/2, π/2]
Tan⁻¹(cot θ) = π/2 − θ, θ∈[0,π]
Cot⁻¹(tan θ) = π/2 − θ, θ∈[−π/2, π/2]
Sec⁻¹(cosec θ) = π/2 − θ, θ∈[−π/2, 0]∪[0, π/2]
Cosec⁻¹(sec θ) = π/2 − θ, θ∈[0,π]−{π/2}
Sin⁻¹(x) = cos⁻¹[√(1−x2)], 0≤x≤1 = −cos⁻¹[√(1−x2)], −1≤x<0

Property Set 5:

Sin⁻¹x + Cos⁻¹x = π/2
Tan⁻¹x + Cot⁻¹(x) = π/2
Sec⁻¹x + Cosec⁻¹x = π/2

Property Set 6:

If x, y > 0

Tan⁻¹x + Tan⁻¹y = π + tan⁻¹ (x+y/ 1-xy), if xy > 1

Tan⁻¹x + Tan⁻¹y = tan⁻¹ (x+y/ 1-xy), if xy < 1

If x, y < 0

Tan⁻¹x + Tan⁻¹y = tan⁻¹ (x+y/ 1-xy), if xy < 1

Tan⁻¹x + Tan⁻¹y = -π + tan⁻¹ (x+y/ 1-xy), if xy > 1

Property Set 7:

sin⁻¹(x) + sin⁻¹(y) = sin⁻¹[x√(1−y2)+ y√(1−x2)]
cos⁻¹x + cos⁻¹y = cos⁻¹[xy−√(1−x2)√(1−y2)]

Property Set 8:

sin⁻¹(sin x) = −π−π, if x∈[−3π/2, −π/2]

= x, if x∈[−π/2, π/2]

= π−x, if x∈[π/2, 3π/2]

=−2π+x, if x∈[3π/2, 5π/2] And so on.

cos⁻¹(cos x) = 2π+x, if x∈[−2π,−π]

= −x, ∈[−π,0]

= x, ∈[0,π]

= 2π−x, ∈[π,2π]

=−2π+x, ∈[2π,3π]

tan⁻¹(tan x) = π+x, x∈(−3π/2, −π/2)

= x, (−π/2, π/2)

= x−π, (π/2, 3π/2)

= x−2π, (3π/2, 5π/2)

Property Set 9: