What is the principal value of (A) $$ \cos^{-1} $$ (1 /√2 ) (B) $$ \tan^{-1}\text{ }(\infty) $$

Let's solve each part of the problem:

### (A) Principal Value of $ \cos^{-1} \left(\frac{1}{\sqrt{2}}\right) $

The inverse cosine function, $ \cos^{-1}(x) $, gives the angle whose cosine is $ x $. The principal value of $ \cos^{-1}(x) $ is restricted to the interval $ [0, \pi] $.

For $ \cos^{-1} \left(\frac{1}{\sqrt{2}}\right) $:

We know that:
\[
\cos \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}
\]

Therefore:
\[
\cos^{-1} \left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4}
\]

So, the principal value is:
\[
\frac{\pi}{4}
\]

### (B) Principal Value of $ \tan^{-1}(\infty) $

The inverse tangent function, $ \tan^{-1}(x) $, gives the angle whose tangent is $ x $. The principal value of $ \tan^{-1}(x) $ is restricted to the interval $ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) $.

For $ \tan^{-1}(\infty) $:

As $ x $ approaches infinity, $ \tan^{-1}(x) $ approaches $ \frac{\pi}{2} $ because the tangent function approaches infinity as the angle approaches $ \frac{\pi}{2} $ from the left.

Therefore:
\[
\tan^{-1}(\infty) = \frac{\pi}{2}
\]

So, the principal value is:
\[
\frac{\pi}{2}
\]

### Final Answers
- **(A)** $ \cos^{-1} \left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4} $
- **(B)** $ \tan^{-1}(\infty) = \frac{\pi}{2} $

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### Related Questions
1. What is the principal value of $ \cos^{-1}(-\frac{1}{\sqrt{2}}) $?
2. How does the range of the inverse trigonometric functions affect the principal value?
3. What is the value of $ \tan^{-1}(1) $?
4. How do you interpret $ \tan^{-1}(-\infty) $?
5. What are the principal values of the inverse sine function?

**Tip:** The principal value of an inverse trigonometric function is the value that lies within its principal range, which is the range of angles where the function is defined to be one-to-one.

Explore the principal values of inverse trigonometric functions, specifically solving $ \cos^{-1}\left(\frac{1}{\sqrt{2}}\right) $ and $ \tan^{-1}(\infty) $. Learn how to determine these values within their restricted intervals and understand their implications.

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