_ x → (π)/(4) co t ^ 3 x - x cos (x + (π)/(4)) is

$= \lim_{x \to \dfrac{\pi}{4}}\dfrac{\left(\dfrac{1}{\tan^{3}x} - \tan x\right)}{\left(\dfrac{1}{\sqrt{2}}\cos x - \dfrac{1}{\sqrt{2}}\sin x\right)}\left[∵ cot\theta = \dfrac{1}{tan\theta}\right]$

$= \lim_{x \to \dfrac{\pi}{4}}\dfrac{\left(1 - ta{n}^{4}x\right)}{ta{n}^{3}x . \left(\dfrac{1}{\sqrt{2}}\cos x - \dfrac{1}{\sqrt{2}}\sin x\right)}$

$= \lim_{x \to \dfrac{\pi}{4}}\dfrac{\left(1 - \tan^{2}x\right)\left(1 + \tan^{2}x\right)}{\tan^{3}x . \left(\dfrac{1}{\sqrt{2}}\cos x - \dfrac{1}{\sqrt{2}}\sin x\right)}$

$= \lim_{x \to \dfrac{\pi}{4}}\dfrac{\sqrt{2}\left[\left(1 - \tan x\right)\left(1 + \tan x\right)\sec^{2}x\right]}{ta{n}^{3}x . \left(\cos x - \sin x\right)}\left[∵ 1 + \tan^{2}x = \sec^{2}x \text{ and } {a}^{2} - {b}^{2} = \left(a - b\right)\left(a + b\right)\right]$

$= \lim_{x \to \dfrac{\pi}{4}}\dfrac{\sqrt{2}\left[\left(1 - \dfrac{\sin x}{\cos x}\right)\left(1 + \tan x\right)\sec^{2}x\right]}{\tan^{3}x . \left(\cos x - \sin x\right)}$

$= \lim_{x \to \dfrac{\pi}{4}}\dfrac{\sqrt{2}\left(\cos x - \sin x\right) . \left(1 + \tan x\right) . se{c}^{2}x}{\cos x . ta{n}^{3}x . \left(\cos x - \sin x\right)}$

$= \lim_{x \to \dfrac{\pi}{4}}\dfrac{\sqrt{2}\left(1 + \tan x\right)}{\cos x . \dfrac{\sin^{3}x}{\cos^{3}x} . \cos^{2}x}$

$= \lim_{x \to \dfrac{\pi}{4}}\dfrac{\sqrt{2}\left(1 + \tan x\right)}{\sin^{3}x}$

$= \dfrac{\sqrt{2}\left(1 + tan\dfrac{\pi}{4}\right)}{\sin^{3}\dfrac{\pi}{4}}$

$= 8$.