Question
Find the value of:
\[ \cos^{-1}(\tan \tfrac{3\pi}{4}) \]
Solution
First, evaluate:
\[ \tan \tfrac{3\pi}{4} = -1 \]
So the expression becomes:
\[ \cos^{-1}(-1) \]
We know that:
\[ \cos \pi = -1 \]
And the principal value range of \( \cos^{-1}x \) is:
\[ [0, \pi] \]
Thus,
\[ \cos^{-1}(-1) = \pi \]
Final Answer:
\[ \boxed{\pi} \]
Key Concept
Always evaluate the inner trigonometric function first, then apply inverse function using principal range.