The principal value of tan^{-1} left( cot fra | vedclass

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(C) We need to find the principal value of $	an^{-1} \left( \cot \frac{43\pi}{4} ight)$.First,simplify the argument of the cotangent function:$\fra

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$-\frac{\pi}{4}$

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(C) We need to find the principal value of $	an^{-1} \left( \cot \frac{43\pi}{4} ight)$.First,simplify the argument of the cotangent function:$\frac{43\pi}{4} = \frac{40\pi + 3\pi}{4} = 10\pi + \frac{3\pi}{4}$.Since $\cot(n\pi + 	heta) = \cot 	heta$ for any integer $n$,we have:$\cot \left( \frac{43\pi}{4} ight) = \cot \left( 10\pi + \frac{3\pi}{4} ight) = \cot \frac{3\pi}{4}$.Now,express $\cot 	heta$ in terms of $	an$ using the identity $\cot 	heta = 	an \left( \frac{\pi}{2} - 	heta ight)$:$\cot \frac{3\pi}{4} = 	an \left( \frac{\pi}{2} - \frac{3\pi}{4} ight) = 	an \left( \frac{2\pi - 3\pi}{4} ight) = 	an \left( -\frac{\pi}{4} ight)$.Thus,$	an^{-1} \left( \cot \frac{43\pi}{4} ight) = 	an^{-1} \left( 	an \left( -\frac{\pi}{4} ight) ight)$.Since $-\frac{\pi}{4}$ lies in the principal value branch of $	an^{-1}x$,which is $(-\frac{\pi}{2}, \frac{\pi}{2})$,the value is $-\frac{\pi}{4}$.

The principal value of $\tan^{-1} \left( \cot \frac{43\pi}{4} \right)$ is

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