Question

1 + cos(2x) = cot(x)sin(2x)

Answer 1

use the identity :
cos(2x) = 2cos^2 (x) - 1

so, the left side becomes
1 + cos(2x) = 1 + 2cos^2 (x) - 1 = 2cos^2 (x)

then use the identity to simplify in right side, they are
cot(x) = cos(x)/sin(x) and
sin(2x) = 2sin(x)cos(x)
therefore, cot(x)sin(2x) = cos(x)/sin(x) * 2sin(x)cos(x) = 2cos^2 (x)

LHS = RHS
QED