Prove
tan70=tan 20 + 2tan 50...
A geometric proof can be achieved by drawing a sketch.
Draw a right-angled triangle, base=x, vertical side=y, hypotenuse=w.
The angle at the lower left corner is 50 degrees, 90 degrees at the lower right,
top angle=40 degrees.
Draw another hypotenuse at 70 degrees from the lower left apex and continue the vertical,
as shown in the attachment.
since w = k, as the top triangle is isosceles.
From Pythagoras' theorem....
therefore....
which is