Write the following in the simplest form: ^ -1 1+x + 1-x 2 ,0<x<1 - Vidyadip

Question

Write the following in the simplest form:
$\sin^{-1}\Big\{\frac{\sqrt{1+\text{x}}+\sqrt{1-\text{x}}}{2}\Big\},0<\text{x}<1$

✓

Answer

Let, $\text{x}=\cos\theta$
Now,
$\sin^{-1}\Big\{\frac{\sqrt{1+\text{x}}+\sqrt{1-\text{x}}}{2}\Big\}$
$=\sin^{-1}\Big\{\frac{\sqrt{1+\cos\theta}+\sqrt{1-\cos\theta}}{2}\Big\}$
$=\sin^{-1}\Bigg\{\frac{\sqrt{2\cos^2\frac{\theta}{2}}+\sqrt{2\sin^2\frac{\theta}{2}}}{2}\Bigg\}$
$=\sin^{-1}\bigg\{\frac{\cos\frac{\theta}{2}+\sin\frac{\theta}{2}}{\sqrt3}\bigg\}$
$=\sin^{-1}\Big\{\frac{1}{\sqrt2}\sin\frac{\theta}{2}+\frac{1}{\sqrt2}\cos\frac{\theta}{2}\Big\}$
$=\sin^{-1}\Big\{\sin\Big(\frac{\theta}{2}+\frac{\pi}{4}\Big)\Big\}$
$=\frac{\theta}{2}+\frac{\pi}{4}$
$=\frac{\cos^{-1}\text{x}}{2}+\frac{\pi}{4}$
$\therefore\ \sin^{-1}\Big\{\frac{\sqrt{1+\text{x}}+\sqrt{1-\text{x}}}{2}\Big\}=\frac{\cos^{-1}\text{x}}{2}+\frac{\pi}{4}$

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 "Solve the Following Question.(3 Marks)" from Inverse Trigonometric Functions→Inverse Trigonometric Functions — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Assume that on an average one telephone number out of 15 called between 2 P.M. and 3 P.M. on week days is busy. What is the probability that if six randomly selected telephone numbers are called, at least 3 of them will be busy?

The probability that a person who undergoes kidney operation will recover is 0.5 . Find the probability that of the six patients who undergo similar operations.
(a) None will recover
(b) Half of them will recover.

For two matrices A and B, $\text{A}=\begin{bmatrix}2&1&3\\4&1&0\end{bmatrix},\text{B}=\begin{bmatrix}1&-1\\0&2\\5&0\end{bmatrix}$ verify that $(AB)^T = B^TA^T$.

Find the value of $\lambda$ so that the following vectors are coplanar:
$\vec{\text{a}}=\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}},\vec{\text{b}}=3\hat{\text{i}}+\lambda\hat{\text{j}}+\hat{\text{k}},\vec{\text{c}}=\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}}$

The foot of the perpendicular drawn from the origin to a plane is M(1, 0, 0). Find the vector equation of the plane.

Diffrentiate the following w.r.t.x

$\frac{\left(x^2+2\right)^4}{\sqrt{x^2+5}}$

Solve the following differential equations:

$y d x+\left(x-y^2\right) d y=0$

A water tank in the form of an inverted cone is being emptied at the rate of 2 cubic feet per second. The height of the cone is 8 feet and the radius is 4 feet. Find the rate of change of the water level when the depth is 6 feet.

Find the value of $\lambda$ so that the following lines are perpendicular to each other.$\frac{\text{x}-5}{5\lambda+2}=\frac{2-\text{y}}{5}=\frac{1-\text{z}}{-1},\frac{\text{x}}{1}=\frac{2\text{y}+1}{4\lambda}=\frac{1-\text{z}}{-3}$

If $f(x)=k x \quad$, for $0<x<2$
$=0$, otherwise,
is the probability density function of a random variable $X$. then find :
(1) Value of $k$,
(ii) $P (1< X <2)$