Write the value of ^ -1 x+ ^ -1 (1 x ) for x > 0. - Vidyadip

Question

Write the value of $\tan^{-1}\text{x}+\tan^{-1}\Big(\frac{1}{\text{x}}\Big)$ for x > 0.

✓

Answer

$\tan^{-1}\text{x}+\tan^{-1}\text{y}=\tan^{-1}\Big(\frac{\text{x}+\text{y}}{1-\text{xy}}\Big),\text{xy}<1$
$\therefore\ \tan^{-1}\text{x}+\tan^{-1}\frac{1}{\text{x}}=\tan^{-1}\bigg(\frac{\text{x}+\frac{1}{\text{x}}}{1-\text{x}\frac{1}{\text{x}}}\bigg),\text{x}>0$
$=\tan^{-1}\Big(\frac{\text{x}^2+1}{0}\Big)$
$=\tan^{-1}(\infty)$
$=\tan^{-1}\Big(\tan\frac{\pi}{2}\Big)$
$=\frac{\pi}{2}$
$\therefore\ \tan^{-1}\text{x}+\tan^{-1}\frac{1}{\text{x}}=\frac{\pi}{2}$

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 "3 Marks Question" 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

Show that the following systems of linear equations has infinite number of solutions and solve:

x - y + 3z = 6,

x + 3y - 3z = -4,

5x + 3y + 3z = 10

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.$\text{f}(\text{x})=\sin\text{x}-\sin2\text{x}-\text{x}\text{ on }[0,\pi]$

Find the value of $\lambda$ so that the following vectors are coplanar:
$\vec{\text{a}}=\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}},\vec{\text{c}}=\lambda\hat{\text{i}}-\hat{\text{j}}+\lambda\hat{\text{k}}$

Evaluate $\int e^{2 x}\left(\frac{1-\sin 2 x}{1-\cos 2 x}\right) d x$

If $\text{y}=\sin\text{x}$ and x changes from $\frac{\pi}{2}$ to $\frac{22}{14}$, what is the approximate change in y?

Show that the function $f(x)$ defined by $f ( x )=\left\{\begin{array}{ll}\frac{\sin x}{x}+\cos x, & x>0 \\ 2, & x=0 \text { is continuous at } x =0 . \\ \frac{4(1-\sqrt{1-x})}{x}, & x<0\end{array}\right.$

Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}=\frac{1-\cos2\text{y}}{1+\cos2\text{y}}$

Find gof and fog when $f : R \rightarrow R$ and $g : R \rightarrow R$ are defined by:
$f(x) = 2x + 3$ and $g(x) = x^2 + 5$

Evaluate $\sin\Big(\frac{1}{2}\sin^{-1}\frac{4}{5}\Big).$

If $\overrightarrow{\text{PQ}}=3\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}}$ and the coordinates of P are (1, -1, 2), find the coordinates of Q.