Evaluate: tan-1(−1) + cos-1(−1/√2)
Solution:
Using principal values:
\[ \tan^{-1}(-1) = -\frac{\pi}{4} \]
(Range of tan-1(x) is \((- \pi/2, \pi/2)\))
\[ \cos^{-1}\left(-\frac{1}{\sqrt{2}}\right) = \frac{3\pi}{4} \]
(Range of cos-1(x) is \([0, \pi]\))
Now,
\[ \tan^{-1}(-1) + \cos^{-1}\left(-\frac{1}{\sqrt{2}}\right) = -\frac{\pi}{4} + \frac{3\pi}{4} \]
\[ = \frac{2\pi}{4} = \frac{\pi}{2} \]
Final Answer:
Value = \[ \frac{\pi}{2} \]