find the exact value of the expression.\n(\tan^{-1}(cot\frac{2pi}{3}))\nselect the correct choice and, if…

find the exact value of the expression.\n(\tan^{-1}(cot\frac{2pi}{3}))\nselect the correct choice and, if necessary, fill in the answer box to complete your choice.\n(\bigcirc) a. (\tan^{-1}(cot\frac{2pi}{3})=)\n(simplify your answer. type an exact answer, using (pi) as needed. use integers or fractions for any numbers in the expression.)\n(\bigcirc) b. there is no solution.

find the exact value of the expression.\n(\tan^{-1}(cot\frac{2pi}{3}))\nselect the correct choice and, if necessary, fill in the answer box to complete your choice.\n(\bigcirc) a. (\tan^{-1}(cot\frac{2pi}{3})=)\n(simplify your answer. type an exact answer, using (pi) as needed. use integers or fractions for any numbers in the expression.)\n(\bigcirc) b. there is no solution.

Answer

Explanation:

Step1: Calculate the value of (\cot\frac{2\pi}{3})

We know that (\cot\theta=\frac{\cos\theta}{\sin\theta}). For (\theta = \frac{2\pi}{3}), (\cos\frac{2\pi}{3}=-\frac{1}{2}) and (\sin\frac{2\pi}{3}=\frac{\sqrt{3}}{2}). So, (\cot\frac{2\pi}{3}=\frac{\cos\frac{2\pi}{3}}{\sin\frac{2\pi}{3}}=\frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}}=-\frac{1}{\sqrt{3}})

Step2: Calculate the value of (\tan^{- 1}\left(-\frac{1}{\sqrt{3}}\right))

Let (y = \tan^{-1}x), where (x\in(-\infty,\infty)) and (y\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)). We know that (\tan\left(-\frac{\pi}{6}\right)=-\frac{1}{\sqrt{3}})

Answer:

(\tan^{-1}\left(\cot\frac{2\pi}{3}\right)=-\frac{\pi}{6})