Solve the equation for x: ^ -1 x + ^ -1 (1 - x) = ^ -1 x - Vidyadip

Question

Solve the equation for $x: \sin^{-1}x + \sin^{-1}(1 - x) = \cos^{-1}x$

✓

Answer

$\sin^{-1}\text{x} + \sin^{-1}(1 - \text{x}) = \cos^{-1}\text{x}\Rightarrow\sin^{-1}(1 - \text{x})= \frac{\pi}{2} - 2\sin^{-1}\text{x}$
$\Rightarrow1 - \text{x} = \sin\bigg(\frac{\pi}{2}-2\sin^{-1}\text{x}\bigg)\Rightarrow1 - \text{x} = \cos(2\sin^{-1}\text{x})\Rightarrow1 - \text{x} = 1 - 2\sin^{2}(\sin^{-1}\text{x})$
$\Rightarrow1 - \text{x} = 1 - \text{2x}^{2}$
Solving we get, $\text{x} =0 \text{ or x} =\frac{1}{2}$

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" from INVERSE TRIGNOMETRIC FUNCTIONS→INVERSE TRIGNOMETRIC FUNCTIONS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Ten cards numbered 1 through 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than 3, what is the probability that it is an even number?

Form the differential equation of the family of curves y = a cos(x + b), where a and b are arbitrary constants.

If $\cos^{-1}\frac{\text{x}}{2}+\cos^{-1}\frac{\text{y}}{3}=\alpha,$ then prove that

$9\text{x}^2-12\text{xy}\cos\alpha+4\text{y}^2=36\sin^2\alpha.$

Prove that $\cos ^{-1} \frac{63}{65}+2 \tan ^{-1} \frac{1}{5}=\sin ^{-1} \frac{3}{5}$.

Let $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}$ and $\vec{\text{c}}=\text{c}_1\hat{\text{i}}+\text{c}_2\hat{\text{j}}+\text{c}_3\hat{\text{k}}.$ Then,
If c₁= 1 and c₂= 2, find c₃which makes $\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ coplanar.

Find the vector equation of the plane with intercepts 3, -4 and 2 on x, y and z-axis respectively.

A pair of dice is thrown. Find the probability of getting 7 as the sum, if it is known that the second die always exhibits an odd number.

Find the coordinates of the point where the line

$\frac{\text{x + 1}}{2}=\frac{\text{y + 2}}{3}=\frac{\text{z + 3}}{4}$

meets the plane x + y + 4z = 6.

Write the value of $\sin^{-1}(\sin(-600^\circ))\sin(-600^\circ).$

Show that the line through points (1, -1, 2) and (3, 4, -2) is perpendicular to the line throught the points (0, 3, 2) and (3, 5, 6).