Question

The value of \( x \) satisfying the equation \( \tan^{-1} x + \tan^{-1} \left(\frac{2}{3}\right) = \tan^{-1} \left(\frac{7}{4}\right) \) is:

• A. $\frac{1}{2}$
• B. $-\frac{1}{2}$
• C. $\frac{3}{2}$
• D. $-\frac{1}{3}$
• E. $\frac{1}{3}$

Hint:

Always check if $AB > -1$ when using the subtraction formula to ensure you are within the principal value range. Here, $(7/4) \times (2/3) = 14/12$, which is greater than $-1$, so the formula is directly applicable.

Answer 1

The Correct Option is A

Concept: To solve for $x$ in an inverse trigonometric equation, we isolate the variable term and use the subtraction formula for inverse tangents. The formula $\tan^{-1} A - \tan^{-1} B = \tan^{-1} \left( \frac{A - B}{1 + AB} \right)$ allows us to merge two inverse tangent terms into one.

Step 1: Isolate the term containing the unknown variable $x$.
Subtract $\tan^{-1} (2/3)$ from both sides of the equation: \[ \tan^{-1} x = \tan^{-1} \left(\frac{7}{4}\right) - \tan^{-1} \left(\frac{2}{3}\right) \]

Step 2: Apply the inverse tangent subtraction formula.
Let $A = 7/4$ and $B = 2/3$. Using the identity: \[ \tan^{-1} x = \tan^{-1} \left( \frac{\frac{7}{4} - \frac{2}{3}}{1 + (\frac{7}{4} \cdot \frac{2}{3})} \right) \] Take the tangent of both sides to remove the inverse function: \[ x = \frac{\frac{7}{4} - \frac{2}{3}}{1 + \frac{14}{12}} \]

Step 3: Perform the fraction arithmetic.
Find a common denominator for the numerator (which is 12): \[ \text{Numerator: } \frac{7 \cdot 3 - 2 \cdot 4}{12} = \frac{21 - 8}{12} = \frac{13}{12} \] Simplify the denominator: \[ \text{Denominator: } 1 + \frac{14}{12} = \frac{12 + 14}{12} = \frac{26}{12} \] Divide the simplified components: \[ x = \frac{13/12}{26/12} = \frac{13}{26} = \frac{1}{2} \]