^ -1 1-a^2 1+a^2 + ^ -1 1-b^2 1+b^2 =2 ^ -1 x - Vidyadip

Question

$\cos ^{-1} \frac{1-a^2}{1+a^2}+\cos ^{-1} \frac{1-b^2}{1+b^2}=2 \tan ^{-1} x$

✓

Answer

Suppose that then$
a =\tan \theta_1, b=\tan \theta_2$
$\theta_1 =\tan ^{-1} a, \theta_2=\tan ^{-1} b$
$\therefore \frac{1-a^2}{1+a^2}=\frac{1-\tan ^2 \theta_1}{1+\tan ^2 \theta_1}=\cos 2 \theta_1$
and $ \frac{1-b^2}{1+b^2}=\frac{1-\tan ^2 \theta_2}{1+\tan ^2 \theta_2}=\cos 2 \theta_2$
hence from given equation :$\cos ^{-1}\left(\cos 2 \theta_1\right)+\cos ^{-1}\left(\cos 2 \theta_2\right)=2 \tan ^{-1} x$
$2 \theta_1+2 \theta_2=2 \tan ^{-1} x$
$\theta_1+\theta_2=\tan ^{-1} x$
$\Rightarrow \tan \left(\theta_1+\theta_2\right)=x$
$\Rightarrow \frac{\tan \theta_1+\tan \theta_2}{1-\tan \theta_1 \tan \theta_2}=x$
$\Rightarrow \frac{a+b}{1-a b}=x$
$\therefore x=\frac{a+b}{1-a b} \text {}$

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 "Answer the question [ 4 mark question ]" from Inverse Trigonometric Functions→Inverse Trigonometric Functions — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Minimize $Z = x +2 y$ subject to constraints
$2 x+2 y \geq 3 ;$
$x+2 y \geq 6 ;$
$x, y \geq 0$
using graphical method.

Find the value of $\int \frac{1}{x\left[6(\log x)^2+7(\log x)+2\right]} d x$

Solve the following LPP using graphical method
$
\begin{array}{ll}
\text { Minimize } & Z=600 x+400 y \\
\text { constraints } & x+2 y>12 \\
& 2 x+y<12 \\
& x+\frac{5}{4} y \geq 5 \\
& x>0, y>0
\end{array}
$

Find the solution of diffrential equation $\left(\tan ^{-1} y-x\right) d y=\left(1+y^2\right) d x$, when $x=0, y=0$.

Solve the following equation $\left(1+y^2\right)+\left(x-e^{\tan ^{-1} y}\right) \frac{d y}{d x}=0$.

If $A=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right]$ and $A^2-4 A=k I_3$ then find the value of $k$. Here $I_3$ has a order of $3 \times 3$ matrix.

Solve the following linear programming problem graphically.
$
\begin{array}{cl}
\operatorname{maximize} & z=20 x+30 y \\
\text { constraints } & x+2 y \leq 20 \\
& 3 x+2 y \leq 30 \\
& x \geq 0, y \geq 0
\end{array}
$

If $A =\left[\begin{array}{ccc}2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0\end{array}\right]$,then find the value of $A ^{ 2 }- 3 A + 2 I$.

If given function is continuous at $x =0$ then write the value of $k$.
$f(x)=\left\{\begin{array}{cl}\frac{\log (1+a x)-\log (1-b x)}{x}, & x \neq 0 \\ k, & x=0\end{array}\right.$

Find the maximum and minimum value of following function. $2 x^3-15 x^2+36 x+10$