Find the principal values of  cot^{-1}(tan\dfrac{3\pi}{4})

But we know that \( tan \dfrac{3\pi}{4} = -1\)

By substituting this value in \( cot^{-1}(tan \dfrac{3\pi}{4})\) we get

Cot^-1(-1)

Now, let y = cot^-1(-1)

Cot y = (-1)

\( – Cot (\dfrac{\pi}{4}) = 1\)

= \( Cot (π – \dfrac{\pi}{4})\)

= \( cot (\dfrac{3\pi}{4})\)

The range of principal value of cot^-1(0, π) and \( cot (\dfrac{3\pi}{4})\) = – 1

Therefore the principal value of \( cot^{-1}(tan \dfrac{3\pi}{4})\) is \( \dfrac{3\pi}{4}\)