Simplify: ( cos 4 ⁡ x − sin 4 ⁡ x ) = 2 sin 2 ⁡ x − 1But how can anyone show why this is true?I ended with sin 2 ⁡ x − cos 2 ⁡ x

Question

Simplify:       (          cos      4        ⁡    x    −          sin      4        ⁡    x    )    =  2      sin    2    ⁡  x  −  1But how can anyone show why this is true?I ended with       sin    2    ⁡  x  −      cos    2    ⁡  x

retalibry9 · Accepted Answer

The given equation is merely conditionally true, but becomes an identity once a negative sign inserted in front:                                  −                  (                      cos            4                    ⁡          x          −                      sin            4                    ⁡          x          )                                    ≡                              sin          4                ⁡        x        −        (        1        −                  sin          2                ⁡        x                  )          2                                    ≡                    2                  sin          2                ⁡        x        −        1

Jaslyn Sloan · Accepted Answer

This statement is false.                              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        −                  sin                      2                          ⁡        x        −                  sin                      2                          ⁡        x                                          =        1        −        2                  sin                      2                          ⁡        x

Simplify (cos^4 x−sin^4 x)=2 sin^2 x−1

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