For x ∈ (-1, 1], the number of solutions of the equation sin⁻¹x = 2tan⁻¹x is equal to _________.

Q: Wait, why can't I just cancel $x$ from both sides of $x(1+x^2) = 2x$?

This is a classic trap! If you divide both sides by $x$, you are implicitly assuming $x \neq 0$. If $x=0$, the equation $0=0$ is true, so $x=0$ is a valid solution. By dividing, you lose this root. Always move all terms to one side and factorize instead: $x(1+x^2) - 2x = 0$ leads to $x(x^2 - 1) = 0$.

Q: Why is $x = -1$ rejected? It satisfies the equation, right?

You are absolutely right that $x = -1$ satisfies the algebraic equation $x^3 - x = 0$. However, look at the domain given in the problem: $x \in (-1, 1]$. The parenthesis at $-1$ means $-1$ is excluded from the domain. In JEE Advanced, domain constraints are non-negotiable. Always check your final roots against the original interval!

Q: Is there a graphical way to see this?

Yes! If you plot $y = \sin^{-1}x$ and $y = 2\tan^{-1}x$, you will see they intersect at three points: $x = -1$, $x = 0$, and $x = 1$. The graph confirms the algebraic roots. The domain constraint then acts as a filter, removing the intersection at $x = -1$.

Question

For $x \in (-1, 1]$, the number of solutions of the equation $\sin^{-1}x = 2	an^{-1}x$ is equal to _________.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow traveler, to the elegant world of inverse trigonometry. Today, we are tackling a problem that seems deceptively simple: finding the number of solutions for $\sin^{-1}x = 2	an^{-1}x$ within the domain $x \in (-1, 1]$.

At first glance, this looks like a straightforward equality, but beneath the surface lies a beautiful interplay of domain constraints and algebraic precision. Let's begin by examining our playground: the interval $x \in (-1, 1]$.

Notice the open parenthesis at $-1$ and the closed bracket at $1$. This is not a mere formality; it is a boundary condition that will dictate our final answer. Many students rush into solving the equation without respecting these walls, but we are not like them. We are precise.

### The Bridge Between Worlds

To solve this, we need a bridge between $\sin^{-1}x$ and $2	an^{-1}x$. We cannot easily compare these two functions directly; we need a common language.

Recall the powerful identity from your trigonometric toolkit:

$$2	an^{-1}x = \sin^{-1}\left(\frac{2x}{1+x^2}ight)$$

This identity is valid for $|x| \le 1$, which perfectly encompasses our given domain. By substituting this into our original equation, we transform the problem into:

$$\sin^{-1}x = \sin^{-1}\left(\frac{2x}{1+x^2}ight)$$

Since the $\sin^{-1}$ function is strictly increasing and one-to-one, we can equate the arguments directly:

$$x = \frac{2x}{1+x^2}$$

### The Algebraic Transformation

Now, the algebra begins. We cross-multiply to get $x(1+x^2) = 2x$. Expanding this, we arrive at $x + x^3 = 2x$.

Rearranging all terms to one side, we get $x^3 - x = 0$. Here is where many students stumble. Do not simply divide by $x$, as you will lose the root $x=0$.

Instead, factorize: $x(x^2 - 1) = 0$, which further breaks down into $x(x-1)(x+1) = 0$. This yields three potential candidates: $x = 0$, $x = 1$, and $x = -1$.

### The Final Filter

We have found three roots, but are they all valid? We must return to our domain constraint: $x \in (-1, 1]$.

We check each root:
*   $x = 0$ is in the interval.
*   $x = 1$ is in the interval (because of the closed bracket).
*   $x = -1$ is rejected because it lies outside the open interval.

Thus, our valid solutions are $x = 0$ and $x = 1$. The **number of solutions is exactly two**.

This problem teaches us that math is not just about finding roots; it is about validating them against the reality of the problem's constraints. Keep this in mind, and you will never be trapped by a domain again.

Animated Solution for Mathematics - Inverse Trigonometric Functions: For $x \in (- 1, 1]$ , the number of solutions of the equation $sin^{- 1} x = 2 tan^{- 1} x$ is equal to _________.

Enter Numerical Value:

Visualized Solution

The Equation and Domain

Inverse Trigonometric Identity

Substituting the Identity

Equating the Arguments

Cross Multiplication

Rearranging into a Polynomial

Factorizing to Find Roots

Checking the Domain Constraint

Final Conclusion

The Sigma Insight: Properties of Inverse Trigonometric Functions

Analyzing the Setup

The Bridge Between Worlds

The Algebraic Transformation

The Final Filter

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Animated Solution for Mathematics - Inverse Trigonometric Functions: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For x∈(−1,1], the number of solutions of the equation sin−1x=2tan−1x is equal to _________.