Hello Alyssa,
Given cos(x), we can first find sin(x). Using the well known identity sin2(x) + cos2(x) = 1, we have
sin2(x) + (21/29)2 = 1
sin2(x) + 441/841 = 1
sin2(x) = 1 - 441/841
sin2(x) = 400/841
sin(x) = ±20/29.
Note that the above gives two possible values for sin(x), but since 0° < x < 90° (x is in the first quadrant), we choose the positive value. That is,
sin(x) = 20/29
Now we see that
tan(x) = sin(x)/cos(x) = [20/29]/[21/29]
tan(x) = 20/21
Now use the double angle formula for tangent.
tan(2x) = 2tan(x)/[1 -tan2(x)]
tan(2x) = 2(20/21)/[1 - (20/21)2]
tan(2x) = (40/41)/[1 - 400/441] = (40/41)/[41/441] = (40/41)•(441/41)
tan(2x) = 17640/1681
Hope that helps! Let me know if you need any further explanation.
William