Question 138821
Note: I'm only manipulating the left side. I'm not touching the right side. I'm only showing it for comparison.



*[Tex \LARGE \sin\left(3\theta\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)] Start with the given equation



*[Tex \LARGE \sin\left(2\theta+\theta\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)] Break up *[Tex \Large 3\theta] to get *[Tex \Large 2\theta+\theta]



*[Tex \LARGE \sin\left(2\theta\right)\cos\left(\theta\right)+\cos\left(2\theta\right)\sin\left(\theta\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)] Expand using the sum-difference formulas



*[Tex \LARGE 2\sin\left(\theta\right)\cos\left(\theta\right)\cos\left(\theta\right)+\left(1-2\sin^2\left(\theta\right)\right)\sin\left(\theta\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)] Replace *[Tex \LARGE \sin\left(2\theta\right)] with *[Tex \LARGE 2\sin\left(\theta\right)\cos\left(\theta\right)]. Replace *[Tex \LARGE \cos\left(2\theta\right)] with *[Tex \LARGE 1-2\sin^2\left(\theta\right)].



*[Tex \LARGE \sin\left(\theta\right)\left(2\cos^2\left(\theta\right)+1-2\sin^2\left(\theta\right)\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)] Factor out *[Tex \LARGE \sin\left(\theta\right)]



*[Tex \LARGE \sin\left(\theta\right)\left(2\cos^2\left(\theta\right)-2\sin^2\left(\theta\right)+1\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)] Rearrange the terms



*[Tex \LARGE \sin\left(\theta\right)\left(2\left(\cos^2\left(\theta\right)-\sin^2\left(\theta\right)\right)+1\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)] Factor out 2



*[Tex \LARGE \sin\left(\theta\right)\left(2\cos\left(2\theta\right)+1\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)] Replace *[Tex \LARGE \cos^2\left(\theta\right)-\sin^2\left(\theta\right)] with *[Tex \LARGE \cos\left(2\theta\right)]



*[Tex \LARGE \sin\left(\theta\right)\left(2\left(1-2\sin^2\left(\theta\right)\right)+1\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)] Replace *[Tex \LARGE \cos\left(2\theta\right)] with *[Tex \LARGE 1-2\sin^2\left(\theta\right)]



*[Tex \LARGE \sin\left(\theta\right)\left(2-4\sin^2\left(\theta\right)+1\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)] Distribute



*[Tex \LARGE \sin\left(\theta\right)\left(3-4\sin^2\left(\theta\right)\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)] Combine like terms



*[Tex \LARGE 3\sin\left(\theta\right)-4\sin^3\left(\theta\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)] Distribute





So we've just proven that 


*[Tex \LARGE \sin\left(3\theta\right) = 3\sin\left(\theta\right) - 4\sin^3\left(\theta\right)] 


is an identity.