Question

$\sec(\cos^{-1}(\frac{2024}{2025}))$ is equal to ________.

• A. $\frac{2024}{2025}$
• B. $\frac{2025}{2024}$
• C. $\frac{1}{2025}$
• D. $\frac{-1}{2025}$
• E. $\frac{-2025}{2024}$

Hint:

$\sec(\cos^{-1} x) = \frac{1}{x}$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
This problem involves evaluating a composite function of a trigonometric function (\(\sec\)) and an inverse trigonometric function (\(\cos^{-1}\)). The key is to understand the relationship between \(\sec\) and \(\cos\).
Step 2: Key Formula or Approach
The secant function is the reciprocal of the cosine function: \(\sec(\theta) = \frac{1}{\cos(\theta)}\).
Let \(\theta = \cos^{-1}(x)\). By definition, this means \(\cos(\theta) = x\).
Therefore, \(\sec(\cos^{-1}(x)) = \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{x}\).
Step 3: Detailed Explanation
1. Let the inner part be an angle \(\theta\).
Let \(\theta = \cos^{-1}\left(\frac{2024}{2025}\right)\).
By the definition of the inverse cosine function, this equation is equivalent to:
\[ \cos(\theta) = \frac{2024}{2025} \] 2. Substitute \(\theta\) back into the original expression.
The expression we need to evaluate is \(\sec(\theta)\).
3. Use the reciprocal identity.
We know that \(\sec(\theta) = \frac{1}{\cos(\theta)}\).
Substitute the value of \(\cos(\theta)\) we found in step 1:
\[ \sec(\theta) = \frac{1}{\frac{2024}{2025}} \] 4. Calculate the final value.
\[ \sec(\theta) = \frac{2025}{2024} \] Alternative method (Right Triangle):
1. Let \(\theta = \cos^{-1}\left(\frac{2024}{2025}\right)\). 2. Since \(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\), we can imagine a right triangle where the adjacent side is 2024 and the hypotenuse is 2025. 3. We need to find \(\sec(\theta)\). 4. \(\sec(\theta) = \frac{\text{Hypotenuse}}{\text{Adjacent}}\). 5. From the triangle, \(\sec(\theta) = \frac{2025}{2024}\). Step 4: Final Answer
The value of the expression is \(\frac{2025}{2024}\).