Question

If $\tan^{-1}x = \tan^{-1}(3) - \frac{\pi}{4}$, then $x$ is equal to:

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. 1
• D. 3
• E. 2

Hint:

Always replace constants like $\pi/4$ with their inverse trig equivalent to match the equation structure.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We need to solve an equation for x involving the inverse tangent function. We will use the formula for the difference of two inverse tangent functions.
Step 2: Key Formula or Approach:
1. Recognize that \( \frac{\pi}{4} = \tan^{-1}(1) \).
2. Use the difference formula for inverse tangent:
\[ \tan^{-1}(A) - \tan^{-1}(B) = \tan^{-1}\left(\frac{A-B}{1+AB}\right) \] Step 3: Detailed Explanation:
The given equation is:
\[ \tan^{-1} x = \tan^{-1}(3) - \frac{\pi}{4} \] First, replace \( \frac{\pi}{4} \) with \( \tan^{-1}(1) \):
\[ \tan^{-1} x = \tan^{-1}(3) - \tan^{-1}(1) \] Now, apply the difference formula to the right side, with \( A=3 \) and \( B=1 \):
\[ \tan^{-1} x = \tan^{-1}\left(\frac{3-1}{1+(3)(1)}\right) \] \[ \tan^{-1} x = \tan^{-1}\left(\frac{2}{1+3}\right) \] \[ \tan^{-1} x = \tan^{-1}\left(\frac{2}{4}\right) \] \[ \tan^{-1} x = \tan^{-1}\left(\frac{1}{2}\right) \] From this, we can conclude that:
\[ x = \frac{1}{2} \] Step 4: Final Answer:
The value of x is \( \frac{1}{2} \).