\(\lim\limits_{x \to \frac{\pi}{4}}\frac{tan^3x-tanx}{cos(x+\frac{\pi}{4})}\)
\(=\lim\limits_{x \to \frac{\pi}{4}}\frac{tanx(tan^2x-1)}{cos(x+\frac{\pi}{4})}\)
\(=\lim\limits_{x \to \frac{\pi}{4}}tanx.\lim\limits_{x \to \frac{\pi}{4}}[\frac{-(1-tan^2\,x)}{cos(x+\frac{\pi}{4})}]\)
\(=1\times[-\lim\limits_{x \to \frac{\pi}{4}}\frac{(1+tan\,x)(1-tan\,x)}{cos(x+\frac{\pi}{4})}]\)
\(=\lim\limits_{x \to \frac{\pi}{4}}(1+tanx)\lim\limits_{x \to \frac{\pi}{4}}[\frac{cos\,x-sin\,x}{cos\,x.cos(x+\frac{\pi}{4})}]\)
\(=-2\sqrt 2\times\) \(\lim\limits_{x \to \frac{\pi}{4}}\frac{cos(x+\frac{\pi}{4})}{cos\,x.cos(x+\frac{\pi}{4})}\)
∵ [cos x − sin x = \(\sqrt{2}\,cos\,(x+\frac{\pi}{4})]\)
