JEE Mains · Maths · STD 12 - 2. inverse trigonometric function
The sum of the infinite series
\(\cot ^{-1}\left(\frac{7}{4}\right)+\cot ^{-1}\left(\frac{19}{4}\right)+\) \(\cot ^{-1}\left(\frac{39}{4}\right)+\cot ^{-1}\left(\frac{67}{4}\right)+\ldots .\) is :-
- A \(\frac{\pi}{2}+\tan ^{-1}\left(\frac{1}{2}\right)\)
- B \(\frac{\pi}{2}-\cot ^{-1}\left(\frac{1}{2}\right)\)
- C \(\frac{\pi}{2}+\cot ^{-1}\left(\frac{1}{2}\right)\)
- D \(\frac{\pi}{2}-\tan ^{-1}\left(\frac{1}{2}\right)\)
Answer & Solution
Correct Answer
(D) \(\frac{\pi}{2}-\tan ^{-1}\left(\frac{1}{2}\right)\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \mathrm{T}_{\mathrm{n}}=\tan ^{-1}\left(\frac{4}{4 \mathrm{n}^2+3}\right) \\ & \mathrm{T}_{\mathrm{n}}=\tan ^{-1}\left(\frac{\left(\mathrm{n}+\frac{1}{2}\right)-\left(\mathrm{n}-\frac{1}{2}\right)}{1+\left(\mathrm{n}+\frac{1}{2}\right)\left(\mathrm{n}-\frac{1}{2…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- The number of ways in which \(21\) identical apples can be distributed among three children such that each child gets at least \(2\) apples, isJEE Mains 2024 Medium
- The maximum slope of the curve \(y=\frac{1}{2} x^{4}-5 x^{3}+18 x^{2}-19 x\) occurs at the pointJEE Mains 2021 Hard
- The value of \(\lim _{x \rightarrow 0}\left(\frac{x}{\sqrt[8]{1-\sin x}-\sqrt[8]{1+\sin x}}\right)\) is equal to:JEE Mains 2021 Hard
- The value of \(\cot ^{-1}\left(\frac{\sqrt{1+\tan ^2(2)}-1}{\tan (2)}\right)-\cot ^{-1}\) \(\left(\frac{\sqrt{1+\tan ^2\left(\frac{1}{2}\right)}+1}{\tan \left(\frac{1}{2}\right)}\right) \text { is equal to }\)JEE Mains 2025 Medium
- If \(\left[ {\begin{array}{*{20}{c}}
1&1\\
0&1
\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}
1&2\\
0&1
\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}
1&3\\
0&1
\end{array}} \right]\,........\left[ {\begin{array}{*{20}{c}}
1&{n - 1}\\
0&1
\end{array}} \right]\, = \,\left[ {\begin{array}{*{20}{c}}
1&{78}\\
0&1
\end{array}} \right]\) then the inverse of \(\left[ {\begin{array}{*{20}{c}}
1&n\\
0&1
\end{array}} \right]\) isJEE Mains 2019 Hard - If the area of the region \(\left\{(\mathrm{x}, \mathrm{y}): 1+\mathrm{x}^2 \leq \mathrm{y} \leq \min \{\mathrm{x}+7,11-3 \mathrm{x}\}\right\}\) is A , then 3 A is equal toJEE Mains 2025 Medium
More PYQs from JEE Mains
- For \(x \in R,x \ne 0\), if \(y(x)\) is a differentiable function such that \(x\int\limits_1^x {y\left( t \right)} dt = \left( {x + 1} \right)\int\limits_1^x {ty\left( t \right)} dt\) , then \(y(x)\) equals (where \(C\) is a constant)JEE Mains 2016 Hard
- If \(x=x(t)\) is the solution of the differential equation \((t+1) d x=\left(2 x+(t+1)^4\right) d t, x(0)=2\), then \(x(1)\) equals ...........JEE Mains 2024 Hard
- If \(A\) and \(B\) are the points of intersection of the circle \(x^2+y^2-8 x=0\) and the hyperbola \(\frac{x^2}{9}-\frac{y^2}{4}=1\) and a point P moves on the line \(2 x-3 y+4=0\), then the centroid of \(\triangle \mathrm{PAB}\) lies on the line :JEE Mains 2025 Hard
- Let \(\mathrm{P}\) be a plane passing through the points \((1,0,1),(1,-2,1)\) and \((0,1,-2)\). Let a vector \(\vec{a}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}\) be such that \(\vec{a}\) is parallel to the plane \(P\), perpendicular to \((\hat{i}+2 \hat{j}+3 \hat{k})\) and \(\overrightarrow{\mathrm{a}} \cdot(\hat{\mathrm{i}}+\hat{\mathrm{j}}+2 \hat{\mathrm{k}})=2\), then \((\alpha-\beta+\gamma)^{2}\) equals \(....\)JEE Mains 2021 Hard
- An online exam is attempted by \(50\) candidates out of which \(20\) are boys. The average marks obtained by boys is \(12\) with a variance \(2 .\) The variance of marks obtained by \(30\) girls is also \(2 .\) The average marks of all \(50\) candidates is \(15 .\) If \(\mu\) is the average marks of girls and \(\sigma^{2}\) is the variance of marks of \(50\) candidates, then \(\mu+\sigma^{2}\) is equal to ...... .JEE Mains 2021 Hard
- A set \(S\) contains \(7\) elements. A non-empty subset \(A\) of \(S\) and an element \(x\) of \(S\) are chosen at random. Then the probability that \(x \in A\) isJEE Mains 2014 Hard