Why is cot ⁡ x cos ⁡ x − csc ⁡ x = − sin ⁡ x true?

Question

Why is   cot  ⁡  x  cos  ⁡  x  −  csc  ⁡  x  =  −  sin  ⁡  x true?

hamputlnf · Accepted Answer

cot        ⁡        x        cos        ⁡        x        −        csc        ⁡        x                            =                  (                                    cos              ⁡              x                                      sin              ⁡              x                                )                cos        ⁡        x        −                  1                      sin            ⁡            x                                                            =                                            cos              2                        ⁡            x            −            1                                sin            ⁡            x                                                            =                              −                          sin              2                        ⁡            x                                sin            ⁡            x                                                            =        −        sin        ⁡        (        x        )        .            Note however that this equality is only true if   x  ≠  n  π for n an integer, since the left hand side is not defined at those points, but the right hand side is defined.

Why is cot x cos x−csc x=−sin x true?

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