prove cot x -tan x = 2 cot 2x

Answer 1

cot x - tan x = 2 cot 2x
I will be solving the left side and make it look like the right side. Note that cot x = cos(x) / sin(x) and tan x = sin(x) / cos(x):
cos(x) / sin(x) - sin(x) / cos(x)
Combining,
cos^2 (x) - sin^2 (x) / cos(x)*sin(x):
Note that the numerator cos^2 (x) - sin^2 (x) = cos (2x),
cos (2x) / cos(x)*sin(x)
And sin(2x) = 2 sin(x)*cos(x):
cos (2x) / (1/2)*sin(2x)
Therefore,
2*cos(2x)/sin(2x)
2*cot(2x)

Hope this helps :3