# Write the principal value of $$cot^{-1} \{ cot \bigg( -\frac{\pi}{4} \bigg) \}$$

Ans:

$\frac{3\pi}{4}$

Principal interval of cot is $$(0.\pi)$$  and $$-\frac{\pi}{4}\notin\:(0,\pi)$$

We know that cot$$-\frac{\pi}{4}=cot(\pi-\frac{\pi}{4})=cot\frac{3\pi}{4}$$

$$\Rightarrow\:cot^{-1}cot-\frac{\pi}{4}=cot^{-1}cot\frac{3\pi}{4}=\frac{3\pi}{4}$$

which is in the principal interval of cot.
edited Mar 19, 2013