find the exact value of the expression: tan⁻¹(tan(10/51π)) =

Explanation:

Step1: Recall the property of inverse - tangent function

The property of the inverse - tangent function $y = \tan^{-1}(x)$ is that $\tan^{-1}(\tan(x))=x$ when $-\frac{\pi}{2}<x<\frac{\pi}{2}$.

Step2: Check if $\frac{10\pi}{51}$ is in the range of $\tan^{-1}(x)$

We know that $-\frac{\pi}{2}\approx - 1.57$ and $\frac{\pi}{2}\approx1.57$, and $\frac{10\pi}{51}\approx\frac{10\times3.14}{51}\approx0.62$. Since $-\frac{\pi}{2}<\frac{10\pi}{51}<\frac{\pi}{2}$.

Answer:

$\frac{10\pi}{51}$