How to prove cos 4 ⁡ x − sin 4 ⁡ x − cos 2 ⁡ x + sin 2 ⁡ x is always 0?I know that the 2nd part is always 1, so I need to prove that the first part also equals 1. So how should I prove it ?

Question

How to prove       cos    4    ⁡  x  −      sin    4    ⁡  x  −      cos    2    ⁡  x  +      sin    2    ⁡  x is always 0?I know that the 2nd part is always 1, so I need to prove that the first part also equals 1. So how should I prove it ?

victorollaa5 · Accepted Answer

Simple: it&#039;s not even true. Unless x is a very specific value.Your other statement, that&#039;s a different story. You can write down      cos    4    ⁡  x  +      sin    4    ⁡  x  =      cos    4    ⁡  x  +      sin    4    ⁡  x  −  2      sin    2    ⁡  x      cos    2    ⁡  x  +  2      sin    2    ⁡  x      cos    2    ⁡  x  =  =  (      cos    2    ⁡  x  +      sin    2    ⁡  x      )    2    −  2      sin    2    ⁡  x      cos    2    ⁡  x  =  =  1  −  2      sin    2    ⁡  x      cos    2    ⁡  x  =      cos    2    ⁡  x  +      sin    2    ⁡  x                              (          1          −          2                      cos            2                    ⁡          x          )                ⏟                    ≠      ±      1      So not even this is true. Nothing&#039;s true here.EDIT:Now it&#039;s obvious.      cos    4    ⁡  x  −      sin    4    ⁡  x  −      cos    2    ⁡  x  +      sin    2    ⁡  x  =  (      cos    2    ⁡  x  −      sin    2    ⁡  x  )                              (                      cos            2                    ⁡          x          +                      sin            2                    ⁡          x          )                ⏟                    1        −      cos    2    ⁡  x  +      sin    2    ⁡  x  =  0

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