Hi. I think that to draw a tangent a curve your point has to actually be on the curve.
I rewrote your function slightly. I used the following code to put a point inside the plot showing where the tangent line passes:
https://mathematica.stackexchange.com/questions/101660/how-to-put-single-points-inside-the-plot
$Version
fun[x_]:=Sin[x]
ptslope[x_,x0_]:= Plot[{fun[x],fun'[x0]*(x-x0)+fun[x0]},{x,x0-2,x0+2},
Epilog -> {Blue, PointSize@Large, Point[{x0, fun[x0]}]}]
ptslope[x,Pi/4]
Edit: See Dr. Singer's very well written post and also note that his solution takes greater care of watching for pathological behavior such as the non existence of a derivative at a particular point.
