Verify the following identity tan3x=3tanx-tan3x1-3tan2x
Step 1: Finding tan2x:
We know that,
tanA+B=tanA+tanB1-tanAtanB
∴tan2x=tanx+x=tanx+tanx1-tanxtanx=2tanx1-tan2x
Step 2: Finding tan3x:
∴tan3x=tan2x+x=tan2x+tanx1-tan2xtanx=2tanx1-tan2x+tanx1-2tanx1-tan2xtanx=2tanx1-tan2x+tanx÷1-2tanx1-tan2xtanx=2tanx+tanx×1-tan2x1-tan2x÷1-2tan2x1-tan2x=2tanx+tanx-tan3x1-tan2x÷1-tan2x-2tan2x1-tan2x=3tanx-tan3x1-tan2x÷1-3tan2x1-tan2x=3tanx-tan3x1-tan2x×1-tan2x1-3tan2x=3tanx-tan3x1-3tan2x
Hence, the above identity is verified.