If `cos^(-1)((1)/(sqrt5))=theta`, then what is the value of `cosec^-1(sqrt5)?`

Options

A. A) `((pi)/(2))+theta`
B. B) `((pi)/(2))-theta`
C. C) `(pi)/(2)`
D. D) `-theta`

Answer

To solve the problem, we start with the given equation:

1. **Given**: \( \cos^{-1}\left(\frac{1}{\sqrt{5}}\right) = \theta \)

This implies that:

\[ \cos \theta = \frac{1}{\sqrt{5}} \]

2. **Using the co-function identity**: We know that:

\[ \cos \theta = \sin\left(\frac{\pi}{2} - \theta\right) \]

Therefore, we can write:

\[ \sin\left(\frac{\pi}{2} - \theta\right) = \frac{1}{\sqrt{5}} \]

3. **Finding the cosecant**: The cosecant function is the reciprocal of the sine function. Thus, we have:

\[ \csc\left(\frac{\pi}{2} - \theta\right) = \frac{1}{\sin\left(\frac{\pi}{2} - \theta\right)} = \frac{1}{\frac{1}{\sqrt{5}}} = \sqrt{5} \]

4. **Finding the inverse cosecant**: Now we need to find \( \csc^{-1}(\sqrt{5}) \). By definition, if \( \csc x = \sqrt{5} \), then:

\[ x = \csc^{-1}(\sqrt{5}) \]

From the previous step, we have:

\[ \csc\left(\frac{\pi}{2} - \theta\right) = \sqrt{5} \]

Therefore:

\[ \csc^{-1}(\sqrt{5}) = \frac{\pi}{2} - \theta \]

5. **Final expression**: Thus, we conclude that:

\[ \csc^{-1}(\sqrt{5}) = \frac{\pi}{2} - \cos^{-1}\left(\frac{1}{\sqrt{5}}\right) \]

This is the required value of \( \csc^{-1}(\sqrt{5}) \).