To find the value of [tex]\(\cos \left[\cos^{-1}\left(\frac{\sqrt{2}}{2}\right) - \frac{\pi}{2}\right]\)[/tex], let's follow the steps methodically:
1. Understand the problem:
- We need to evaluate the expression [tex]\(\cos \left[\cos^{-1}\left(\frac{\sqrt{2}}{2}\right) - \frac{\pi}{2}\right]\)[/tex].
2. Evaluate the inner expression:
- [tex]\(\cos^{-1}\left(\frac{\sqrt{2}}{2}\right)\)[/tex] is the angle whose cosine is [tex]\(\frac{\sqrt{2}}{2}\)[/tex].
- The angle whose cosine is [tex]\(\frac{\sqrt{2}}{2}\)[/tex] is [tex]\(\frac{\pi}{4}\)[/tex] (or 45 degrees).
3. Substitute the known value:
- Replace [tex]\(\cos^{-1}\left(\frac{\sqrt{2}}{2}\right)\)[/tex] with [tex]\(\frac{\pi}{4}\)[/tex]:
[tex]\[
\cos \left[\frac{\pi}{4} - \frac{\pi}{2}\right]
\][/tex]
4. Simplify inside the cosine function:
- [tex]\(\frac{\pi}{4} - \frac{\pi}{2}\)[/tex] simplifies to:
[tex]\[
\frac{\pi}{4} - \frac{2\pi}{4} = \frac{\pi}{4} - \frac{\pi}{2} = \frac{\pi}{4} - \frac{2\pi}{4} = -\frac{\pi}{4}
\][/tex]
5. Evaluate the cosine of the simplified angle:
- We now need to find [tex]\(\cos\left(-\frac{\pi}{4}\right)\)[/tex].
- The cosine function is even, which means [tex]\(\cos(-x) = \cos(x)\)[/tex]. Thus:
[tex]\[
\cos\left(-\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right)
\][/tex]
6. Know the fundamental value:
- [tex]\(\cos\left(\frac{\pi}{4}\right)\)[/tex] is well known to be:
[tex]\[
\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\][/tex]
Therefore, the value of [tex]\(\cos \left[\cos^{-1}\left(\frac{\sqrt{2}}{2}\right) - \frac{\pi}{2}\right]\)[/tex] is [tex]\(\frac{\sqrt{2}}{2}\)[/tex].
The correct answer is:
d. [tex]\(\frac{\sqrt{2}}{2}\)[/tex]
