Fourier transform of a triangular pulse

Sinc function is tricky, because there are two of them. It seems your book uses the convention $$\operatorname{sinc} x = \frac{\sin (\pi x)}{\pi x}$$ The desired answer is $$X(\tau) = \tau\frac{\sin^2 (\omega \tau/2)}{(\omega \tau/2)^2} = \frac{4}{\omega^2 \tau }\sin^2 (\omega \tau/2) =\frac{2}{\omega^2 \tau }(1-\cos \omega \tau) $$ Which is what you have, since $e^{j\omega\tau}+e^{-j\omega\tau}=2\cos \omega\tau$.