The value of k which the function f(x) = \ cases ((4)/(5))( 4x)/(...

Given function is continuous at $x = \dfrac{\pi}{2}$

$⇒ f\left(\dfrac{\pi}{2}\right) = f\left(\dfrac{{\pi}^{+}}{2}\right) = f\left(\dfrac{{\pi}^{-}}{2}\right)$

$⇒ k + \dfrac{2}{5} = \left(\dfrac{4}{5}\right)\dfrac{\tan 4x}{\tan 5x}$

$⇒ k + \dfrac{2}{5} = {\left(\dfrac{4}{5}\right)}^{\tan \left(2\pi\right)\cot \left(\dfrac{5\pi}{2}\right)}$

$⇒ k + \dfrac{2}{5} = {\left(\dfrac{4}{5}\right)}^{0}$

$⇒ k + \dfrac{2}{5} = 1$

$⇒ k = \dfrac{3}{5}$