Try to write everything in terms of sines and cosines:
1 / cos B - cos B = (sin B / cos B) sin B
1 / cos B - cos B = sin2B / cos B
Multiply by the common denominator, cos B:
1 - cos2B = sin2B
Use the pithagorean identity on the left side:
sin2B + cos2B - cos2B = sin2B
sin2B = sin2B