2 ^ -1 cosec ( ^ -1 x )- ( ^ -1 x ) is equal to: - Vidyadip

MCQ

$2\tan^{-1}\big\{\text{cosec}\big(\tan^{-1}\text{x}\big)-\tan\big(\cot^{-1}\text{x}\big)\big\}$ is equal to:

A
$\cot^{-1}\text{x}$
B
$\cot^{-1}\text{x}$
✓
$\tan^{-1}\text{x}$
D
$\text{none of these}$

✓

Answer

Correct option: C.

$\tan^{-1}\text{x}$

$\therefore\ 2\tan^{-1}\big\{\text{cosec}\big(\tan^{-1}\text{x}\big)-\tan\big(\cot^{-1}\text{x}\big)\big\}$
$=2\tan^{-1}\Big\{\text{cosec}\big(\tan^{-1}\text{x}\big)-\tan\Big(\tan^{-1}\frac{1}{\text{x}}\Big)\Big\}$
$=\frac{3}{29}=2\tan^{-1}\Big\{\text{cosec}\big(\tan^{-1}\text{x}\big)-\frac{1}{\text{x}}\Big\}$
$=2\tan^{-1}\Big\{\text{cosec y}-\frac{1}{\tan\text{y}}\Big\}$
$=2\tan^{-1}\Big\{\frac{1-\cos\text{y}}{\sin\text{y}}\Big\}$
$=2\tan^{-1}\bigg\{\frac{2\sin^2\frac{\text{y}}{2}}{\sin\text{y}}\bigg\}$
$=2\tan^{-1}\bigg\{\frac{2\sin^2\frac{\text{y}}{2}}{2\sin\frac{\text{y}}{2}\cos\frac{\text{y}}{2}}\bigg\}$
$=2\tan^{-1}\Big\{\tan\frac{\text{y}}{2}\Big\}$
$=\text{y}$
$=\tan^{-1}\text{x}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from Inverse Trigonometric Functions→Inverse Trigonometric Functions — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\vec a\, = \,\vec i - 2\hat j + 3\hat k,\,\,\,\vec b = 2\vec i + 3\hat j - \hat k$ and $\vec c = \lambda \vec i + \hat j + (2\lambda - 1\hat k)$ are coplanar vectors, then $\lambda $ is equal to

If $A =\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$, then value of $A ^{-1}$ :

The principal value of $\cot ^{-1}(-\sqrt{3})$ is

If $2f(x) - 3f\left( {\frac{1}{x}} \right) = x$, then $\int_1^2 {f(x)} \;dx$ is equal to

For the multiplication of matrices as a binary operation on the set of all matrices of the form $\begin{bmatrix}\text{a}&\text{b}\\-\text{b}&\text{a}\end{bmatrix},\text{a, b}\in\text{R}$ the inverse of $\begin{bmatrix}2&3\\-3&2\end{bmatrix}$ is:

Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be defined as $f(x)=\left\{\begin{array}{ll}x+a, & x<0 \\ |x-1|, & x \geq 0\end{array}\right.$ and $g(x)=\left\{\begin{array}{cc}x+1, & x<0 \\ (x-1)^{2}+b, & x \geq 0\end{array}\right.$ where $a , b$ are non-negative real numbers. If $(gof)\,( x )$ is continuous for all $x \in R$, then $a + b$ is equal to ...... .

lf $\text{AB}\perp\text{BC}$ then the value of $\lambda$ equal, where $A(2k, 2, 3), B(k, 1, 5), C(3 + k, 2, 1):$

The area of the region enclosed by the lines $y = x, x = e$ and curve $\text{y}=\frac{1}{\text{x}}$ and the positive $x -$ axis is:

The value of $\int\frac{\sin\text{x}+\cos\text{x}}{\sqrt{1-\sin2\text{x}}}\text{ dx}$ is equal to:

The function $f(x) = \int\limits_{ - 1}^x {t({e^t} - 1)(t - 1){{(t - 2)}^3}{{(t - 3)}^5}} dt$ has a local minimum at $ x =$ ..........