Easy Fourier Transform

Question

I am asked to find the fourier transform of $f(x)= a-|x|$ when $x<|a|$ and $f(x)=0$ otherwise. I have done the calculation and end up with $\dfrac{1-ik-e^{-ika}}{k^2}$, the answer shown is $\dfrac{4 \sin^{2}(\frac{ka}{2})}{k^2}$. I get this if i take the real part but that's kind of cheating. Why does $i \sin(ka) - ika = 0$. I feel like I'm missing some simple trig identity, so sorry if this is really stupid.

Answer 1

HINT The function $f(x)$ is a triangular function and can be seen as the convolution of two rectangual functions and then the Fourier transform is the product of two sinc functions.