Question

Verify the identity (-1 + cot^2 x)/ (cot (2x)) = 2 cot x

Answer 1

Start with:
cot(2x) = (cot^2(x) - 1) / (2cot(x))
cot(2x) = (cot^2(x)/2cot(x)) - 1/(2cot(x))
cot(2x) = cot(x)/2 - 1/(2cot(x))

Change to terms of sin and cos:
cos(2x)/sin(2x) = (1/2)*cos(x)/sin(x) - (1/2)*sin(x)/cos(x)
2cos(2x)/sin(2x) = cos(x)/sin(x) - sin(x)/cos(x)
2cos(2x)/sin(2x) = (cos^2(x) - sin^2(x))/(sin(x)cos(x))

Apply double angle formula for sin(2x)
2cos(2x)/[2sin(x)cos(x)] = (cos^2(x) - sin^2(x))/(sin(x)cos(x))
cos(2x)/(sin(x)cos(x)) = (cos^2(x) - sin^2(x))/(sin(x)cos(x))
cos(2x) = (cos^2(x) - sin^2(x))

Apply double angle formula for cos(2x):
cos^2(x) - sin^2(x) = cos^2(x) - sin^2(x)

Since the two sides are equal, the proof is complete.

Answer 2

Thank you.