How to do the following trigonometric simplification: 1 − cos ⁡ ( 3 α ) 1 − cos ⁡ ( α ) = ( 1 + 2 cos ⁡ ( α ) 2 )

Question

How to do the following trigonometric simplification:             1      −      cos      ⁡      (      3      α      )              1      −      cos      ⁡      (      α      )        =  (  1  +  2  cos  ⁡  (  α      )    2    )

luluna81mxmbk · Accepted Answer

I prefer using capital AWe will prove that  1  −  cos  ⁡  3  A  =  (  1  −  cos  ⁡  A  )  (  1  +  2  cos  ⁡  A      )    2  First we will show that  cos  ⁡  3  A  =  4      cos    3    ⁡  A  −  3  cos  ⁡  AIndeed, we know that  cos  ⁡  (  x  +  y  )  =  cos  ⁡  x  cos  ⁡  y  −  sin  ⁡  x  sin  ⁡  yThus, for x=2A,y=A we obtain:  (  ∗  )  :  cos  ⁡  3  A  =  cos  ⁡  2  A  cos  ⁡  A  −  sin  ⁡  2  A  sin  ⁡  Abut from (∗) we see that:   cos  ⁡  2  A  =      cos    2    ⁡  A  −      sin    2    ⁡  Aand we know that   sin  ⁡  2  A  =  2  sin  ⁡  A  cos  ⁡  ASo,  cos  ⁡  3  A  =  (      cos    2    ⁡  A  −      sin    2    ⁡  A  )  cos  ⁡  A  −  2      sin    2    ⁡  A  cos  ⁡  A    =  (  2      cos    2    ⁡  A  −  1  )  cos  ⁡  A  −  2  (  1  −      cos    2    )  cos  ⁡  A    =  4      cos    3    ⁡  A  −  3  cos  ⁡  ANow we have  1  −  cos  ⁡  3  A  =  1  −  4      cos    3    ⁡  A  +  3  cos  ⁡  ABut  (  1  −  cos  ⁡  A  )  (  1  +  2  cos  ⁡  A      )    2    =    (  1  −  cos  ⁡  A  )  (      cos    2    ⁡  A  +  2  cos  ⁡  A  +  1  )  =    1  −  4      cos    3    ⁡  A  +  3  cos  ⁡  Aand we are done.

How to do the following trigonometric simplification: (1-cos(3 alpha))/(1−cos(alpha))=(1+2 cos(alpha)^2)

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