Question

If $\tan^{-1}(x + 1) + \tan^{-1} x + \tan^{-1}(x - 1) = \tan^{-1} 3$, then for $x < 0$ the value of $500x^4 + 270x^2 + 997 = \dots$

• A. 6716
• B. 1767
• C. 1768
• D. 6717

Hint:

When an inverse trig equation evaluates to a polynomial containing only even powers of $x$ (like $x^4$ and $x^2$), it is a massive hint that $x^2$ is an integer (most commonly 1). Testing $x = \pm 1$ early can save immense calculation time.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given an inverse trigonometric identity involving $x$. We must solve for $x$ (specifically a negative value, $x < 0$) and then evaluate the given polynomial expression.

Step 2: Key Formula or Approach:
Use the standard inverse tangent addition formula on the outer two terms first:
$\tan^{-1} A + \tan^{-1} B = \tan^{-1} \left( \frac{A + B}{1 - AB} \right)$
Alternatively, applying the tangent function to both sides of the full equation will yield a cubic polynomial in $x$.

Step 3: Detailed Explanation:
Let's apply the $\tan$ function to the equation, using the formula $\tan(\alpha + \beta + \gamma) = \frac{S_1 - S_3}{1 - S_2}$, where $S_1$ is the sum of terms, $S_2$ is the sum of product pairs, and $S_3$ is the product of all three terms.
Let the arguments be $a = (x+1), b = x, c = (x-1)$.
$$S_1 = (x+1) + x + (x-1) = 3x$$
$$S_2 = (x+1)(x) + x(x-1) + (x+1)(x-1) = (x^2+x) + (x^2-x) + (x^2-1) = 3x^2 - 1$$
$$S_3 = (x+1)(x)(x-1) = x(x^2 - 1) = x^3 - x$$
Applying tangent to both sides of the given equation:
$$\frac{3x - (x^3 - x)}{1 - (3x^2 - 1)} = 3$$
$$\frac{4x - x^3}{2 - 3x^2} = 3$$
Cross-multiply to form a cubic equation:
$$4x - x^3 = 3(2 - 3x^2) = 6 - 9x^2$$
$$-x^3 + 9x^2 + 4x - 6 = 0 \implies x^3 - 9x^2 - 4x + 6 = 0$$
By rational root theorem or simple inspection, test small integer values. Since $x < 0$, let's test $x = -1$:
$$(-1)^3 - 9(-1)^2 - 4(-1) + 6 = -1 - 9 + 4 + 6 = 0$$
Thus, $x = -1$ is a perfect root and satisfies the condition $x < 0$.
Now substitute $x = -1$ into the required polynomial:
$$500(-1)^4 + 270(-1)^2 + 997 = 500(1) + 270(1) + 997$$
$$= 500 + 270 + 997 = 770 + 997 = 1767$$

Step 4: Final Answer:
The polynomial evaluates to 1767, matching option (b).