JEE Mathematics 2023 Question with Solution

Given:

$$2 \tan^{-1}\left(\frac{1-x}{1+x}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right), \quad 0 < x < 1$$

Find: The number of elements in $S$ and the correct option.

Let

$$\tan^{-1}\left(\frac{1-x}{1+x}\right) = \theta$$

Then

$$\frac{1-x}{1+x} = \tan \theta, \quad \theta \in \left(0, \frac{\pi}{4}\right)$$

Using

$$\cos 2\theta = \frac{1-\tan^2\theta}{1+\tan^2\theta}$$

with

$$\tan \theta = \frac{1-x}{1+x}$$

we get

$$\cos 2\theta = \frac{1-x^2}{1+x^2}$$

So the given equation becomes

$$2\theta = \cos^{-1}(\cos 2\theta)$$

Since $\theta \in \left(0, \frac{\pi}{4}\right)$, we have $2\theta \in \left(0, \frac{\pi}{2}\right) \subset [0,\pi]$. Therefore,

$$\cos^{-1}(\cos 2\theta) = 2\theta$$

Hence the equation is satisfied for every $x \in (0,1)$.

This means $S = (0,1)$, so $S$ has infinitely many elements. Therefore the conclusion shown in the solution is inconsistent with the actual equation, and none of the listed options matches the correct mathematical result.

However, the solution explicitly states "The Correct Option is B". the answer is marked as B.

Therefore, the correct option according to the solution is B.

Given:

$$2 \tan^{-1}\left(\frac{1-x}{1+x}\right) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right), \quad 0 < x < 1$$

Find: Whether the options are consistent.

Set

$$t = \frac{1-x}{1+x}$$

For $0 < x < 1$, we get $0 < t < 1$. Let

$$\theta = \tan^{-1}(t)$$

Then $\theta \in \left(0, \frac{\pi}{4}\right)$ and

$$t = \tan \theta$$

Now,

$$\frac{1-t^2}{1+t^2} = \cos 2\theta$$

But

$$t = \frac{1-x}{1+x}$$

so

$$\frac{1-t^2}{1+t^2} = \frac{1-\left(\frac{1-x}{1+x}\right)^2}{1+\left(\frac{1-x}{1+x}\right)^2} = \frac{1-x^2}{1+x^2}$$

Hence

$$\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) = \cos^{-1}(\cos 2\theta)$$

Since $2\theta \in \left(0, \frac{\pi}{2}\right)$, the principal value gives

$$\cos^{-1}(\cos 2\theta) = 2\theta$$

Thus the equation holds identically for all $x \in (0,1)$.

So mathematically,

$$S = (0,1)$$

and therefore $n(S)$ is not $0$, $1$, or $2$. The options are inconsistent with the equation as written.

The extracted the solution nevertheless declares B. This is likely a source mismatch or solution error, but under the extraction rule that the solution is the primary source, the recorded answer is B.