Prove that: 3 ^ - 1 x = ^ - 1 ( 3x - 4 x^3 ),x [ - 1 2 ,1 2 ] - Vidyadip

Question

Prove that: $3{\sin ^{ - 1}}x = {\sin ^{ - 1}}\left( {3x - 4{x^3}} \right),x \in \left[ { - \frac{1}{2},\frac{1}{2}} \right]$

✓

Answer

We know that: $\sin 3\theta = 3\sin \theta - 4{\sin ^3}\theta$
Putting $\sin \theta = x$
$ \Rightarrow \theta = {\sin ^{ - 1}}x$
$\therefore \sin 3\theta = 3\sin \theta - 4{\sin ^3}\theta$
$\Rightarrow \sin 3\theta = 3x - 4{x^3}$
$\Rightarrow 3\theta = {\sin ^{ - 1}}\left( {3x - 4{x^3}} \right)$
Putting $\theta = {\sin ^{ - 1}}x$,
$3{\sin ^{ - 1}}x = {\sin ^{ - 1}}\left( {3x - 4{x^3}} \right)$
Hence Proved.

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 "2 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

Find which of the binary operations are commutative and which are associative.
State whether the following statements are true or false. Justify
If * is a commutative binary operation on N, then a * (b * c) = (c * b) * a

Form the differential equation representing the family of curves y = e^2x(a + bx), where ‘a’ and ‘b’ are arbitrary constants.

If matrix A = [1 2 3], write AA^T.

When three dice are thrown, write the probability of getting 4 or 5 on each of the dice simultaneously.

Evaluate the following integrals:
$\int^\limits{2\pi}_{0}|\sin\text{x}|\text{dx}$

In the following matrix equation use elementary operation R₂ → R₂ + R₁ and the equation thus obtained:
$\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 2 & -1 \end{bmatrix}\begin{bmatrix} 8 & -3 \\ 9 & -4 \end{bmatrix}$

Compute $\left[\begin{array}{cc} {a} & {b} \\ {-b} & {a} \end{array}\right]+\left[\begin{array}{ll} {a} & {b} \\ {b} & {a} \end{array}\right]$

Prove that the identity function on real numbers given by f(x) = x is continuous at every real number.

The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs.
not more than one.

If A and B are square matrices of the same order, explain, why in general:
(A − B)² ≠ A² − 2AB + B²