Question

If $2\cot^{-1}(\frac{4}{3}) = \cos^{-1}(\frac{x}{5})$, then the value of $x$ is equal to

• A. $\frac{3}{25}$
• B. $\frac{7}{25}$
• C. $\frac{3}{5}$
• D. $\frac{7}{5}$
• E. $\frac{5}{7}$

Hint:

Converting all inverse ratios to $\tan^{-1}$ simplifies using the standard double-angle and sum formulas for trigonometric functions.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We can convert the cotangent inverse to tangent inverse and use the double-angle formula for cosine in terms of tangent.
Key Formula or Approach:
\[ \cos 2\theta = \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} \]

Step 2: Detailed Explanation:
1. Let $\cot^{-1}(\frac{4}{3}) = \theta \implies \cot \theta = \frac{4}{3} \implies \tan \theta = \frac{3}{4}$.
2. The given equation is $2\theta = \cos^{-1}(\frac{x}{5})$.
3. This implies $\cos 2\theta = \frac{x}{5}$.
4. Use the formula for $\cos 2\theta$ in terms of $\tan \theta$:
\[ \frac{x}{5} = \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} \]
5. Substitute $\tan \theta = 3/4$:
\[ \frac{x}{5} = \frac{1 - (3/4)^2}{1 + (3/4)^2} = \frac{1 - 9/16}{1 + 9/16} \]
\[ \frac{x}{5} = \frac{(16 - 9)/16}{(16 + 9)/16} = \frac{7}{25} \]
6. Solve for $x$:
\[ x = 5 \cdot \frac{7}{25} = \frac{7}{5} \]

Step 3: Final Answer:
The value of $x$ is $\frac{7}{5}$.