Etydhv

I'm trying to graph the tangent plane on the unit sphere at the point (1, pi/3,pi/4). I'm new to tikz, so I'm unsure how to edit the graphs other users have made with the planes for each point. Here's the current graph of a sphere I have:

begintikzpicture
 beginaxis[%
 axis equal,
 width=10cm,
 height=10cm,
 axis lines = center,
 xlabel = $x$,
 ylabel = $y$,
 zlabel = $z$,
 ticks=none,
 enlargelimits=0.3,
 view/h=45,
 scale uniformly strategy=units only,
 ]
 addplot3[%
 opacity = 0.5,
 surf,
 z buffer = sort,
 samples = 21,
 variable = u,
 variable y = v,
 domain = 0:180,
 y domain = 0:360,
 ]
 (cos(u)*sin(v), sin(u)*sin(v), cos(v));
 endaxis
endtikzpicture

What's the easiest way to add the tangent plane to the graph? Do I have to create 4 nodes and then fill the rectangle they make? Or is it possible to create a plane based on the span of two vectors originating from that point? Any and all help/explanations would be greatly appreciated!

This is a style that inserts a plane at a point that is spanned by two vectors.

documentclass[tikz,border=3.14mm]standalone
usepackagepgfplots
pgfplotssetcompat=1.16
usetikzlibrarycalc
begindocument
defVecAdd(#1,#2,#3)+(#4,#5,#6)->#7XXXedef#7(#1+1*(#4),#2+1*(#5),#3+(#6))
begintikzpicture[tangent plane/.style args=at #1 with vectors #2 and #3%
/utils/exec=edeftempnoexpandVecAdd#1+#2->noexpandmyvecOne XXX%
temp%
edeftempnoexpandVecAddmyvecOne+#3->noexpandmyvecTwo XXX%
temp
edeftempnoexpandVecAdd#1+#3->noexpandmyvecThree XXX%
temp,
insert path=#1 -- myvecOne -- myvecTwo -- myvecThree -- cycle]
 beginaxis[%
 axis equal,
 width=10cm,
 height=10cm,
 axis lines = center,
 xlabel = $x$,
 ylabel = $y$,
 zlabel = $z$,
 ticks=none,
 enlargelimits=0.3,
 view/h=45,
 scale uniformly strategy=units only,
 ]
 addplot3[%
 opacity = 0.5,
 surf,
 z buffer = sort,
 samples = 21,
 variable = u,
 variable y = v,
 domain = 0:180,
 y domain = 0:360,
 ]
 (cos(u)*sin(v), sin(u)*sin(v), cos(v));
 draw[fill=white,fill opacity=0.4,
 tangent plane=at (-0.5,-0.5,1) with vectors (1,0,0) and (0,1,0)];
 endaxis
endtikzpicture
enddocument

The mandatory animation:

documentclass[tikz,border=3.14mm]standalone
usepackagepgfplots
pgfplotssetcompat=1.16
usetikzlibrarycalc
begindocument
defVecAdd(#1,#2,#3)+(#4,#5,#6)->#7XXXedef#7(#1+1*(#4),#2+1*(#5),#3+(#6))
tikzsettangent plane/.style args=at #1 with vectors #2 and #3%
/utils/exec=edeftempnoexpandVecAdd#1+#2->noexpandmyvecOne XXX%
temp%
edeftempnoexpandVecAddmyvecOne+#3->noexpandmyvecTwo XXX%
temp
edeftempnoexpandVecAdd#1+#3->noexpandmyvecThree XXX%
temp,
insert path=#1 -- myvecOne -- myvecTwo -- myvecThree -- cycle
foreach Angle in 90,85,...,0
begintikzpicture
 beginaxis[hide axis,
 axis equal,
 width=10cm,
 height=10cm,
 axis lines = center,
 xlabel = $x$,
 ylabel = $y$,
 zlabel = $z$,
 ticks=none,
 enlargelimits=0.3,
 view/h=45,
 scale uniformly strategy=units only,
 ]
 addplot3[%
 opacity = 0.5,
 surf,
 z buffer = sort,
 samples = 21,
 variable = u,
 variable y = v,
 domain = 0:180,
 y domain = 0:360,
 ]
 (cos(u)*sin(v), sin(u)*sin(v), cos(v));
 draw[overlay,fill=white,fill opacity=0.4,
 tangent plane=at (cos(Angle)-0.5*sin(Angle),-0.5,sin(Angle)) with 
 vectors (sin(Angle),0,-cos(Angle)) and (0,1,0)];
 endaxis
endtikzpicture
enddocument