What is the value of `sin^(-1)""(4)/(5)+sec^(-1)""(5)/(4)-(pi)/(2)`

Options

A. `(pi)/(4)`
B. `(pi)/(2)`
C. `pi`
D. 0

Answer

To solve the expression \( \sin^{-1}\left(\frac{4}{5}\right) + \sec^{-1}\left(\frac{5}{4}\right) - \frac{\pi}{2} \), we can follow these steps:

### Step 1: Rewrite the secant inverse
Recall that \( \sec^{-1}(x) \) can be rewritten in terms of cosine inverse:
\[ \sec^{-1}(x) = \cos^{-1}\left(\frac{1}{x}\right) \]
Thus,
\[ \sec^{-1}\left(\frac{5}{4}\right) = \cos^{-1}\left(\frac{4}{5}\right) \]

### Step 2: Substitute the rewritten secant inverse into the expression
Now we can substitute this back into the original expression:
\[ \sin^{-1}\left(\frac{4}{5}\right) + \cos^{-1}\left(\frac{4}{5}\right) - \frac{\pi}{2} \]

### Step 3: Use the identity for sine and cosine inverse
We know from trigonometric identities that:
\[ \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2} \]
Applying this identity, we have:
\[ \sin^{-1}\left(\frac{4}{5}\right) + \cos^{-1}\left(\frac{4}{5}\right) = \frac{\pi}{2} \]

### Step 4: Substitute back into the expression
Now substituting back into our expression:
\[ \frac{\pi}{2} - \frac{\pi}{2} = 0 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{0} \]