Question

Prove that tan3x/tanx = 2cos2x+1/2cos2x-1

Answer 1

tan3x/tanx
=(sin3x/cos3x)/(sinx/cosx)
={(3sinx-4sin³x)/(4cos³x-3cosx)}/(sinx/cosx)
={sinx(3-4sin²x)/cosx(4cos²x-3)}×(cosx/sinx)
=(3-4sin²x)/(4cos²x-3)
={3-4(1-cos²x)}/{2(2cos²x)-3}
=(3-4+4cos²x)/{2(1+cos2x)-3}
=(4cos²x-1)/(2+2cos2x-3)
={2(2cos²x)-1}/(2cos2x-1)
={2(1+cos2x)-1}/(2cos2x-1)
=(2+2cos2x-1)/(2cos2x-1)
=(2cos2x+1)/(2cos2x-1) (Proved)