Simplifying ω sin ⁡ ( ω t ) tan ⁡ ( ω t ) + ω ( cos ⁡ ( ω t ) + 1 ) sec 2 ⁡ ( ω t ) ( cos ⁡ ( ω t ) + 1 ) 2

Question

Simplifying             ω      sin      ⁡      (      ω      t      )      tan      ⁡      (      ω              t            )      +      ω      (      cos      ⁡      (      ω      t      )      +      1      )              sec        2            ⁡      (      ω      t      )              (      cos      ⁡      (      ω      t      )      +      1              )        2

Stella Andrade · Accepted Answer

ω        sin        ⁡                  (          ω          t          )                tan        ⁡                  (          ω                      t                    )                +        ω                  (          cos          ⁡                      (            ω            t            )                    +          1          )                          sec                      2                          ⁡                  (          ω          t          )                                      (          cos          ⁡                      (            ω            t            )                    +          1          )                          2                      ,    sec  ⁡  (  ω      t    )  =      1          cos      ⁡      (      ω              t            )        ,    tan  ⁡  (  ω      t    )  =            sin      ⁡      (      ω              t            )              cos      ⁡      (      ω              t            )        ,    ⟹        ⟹                      ω        sin        ⁡                  (          ω          t          )                tan        ⁡                  (          ω                      t                    )                +        ω                  (          cos          ⁡                      (            ω            t            )                    +          1          )                          sec                      2                          ⁡                  (          ω          t          )                                      (          cos          ⁡                      (            ω            t            )                    +          1          )                          2                      =            ω                                    sin                          2                                ⁡          (          ω                      t                    )                          cos          ⁡          (          ω                      t                    )                    +      ω              (                              cos            ⁡            (            ω                          t                        )            +            1                                              cos                              2                                      ⁡            (            ω                          t                        )                          )                            (        cos        ⁡                  (          ω                      t                    )                +        1        )                    2              =    =            ω                                    sin                          2                                ⁡          (          ω                      t                    )          cos          ⁡          (          ω                      t                    )                                      cos                          2                                ⁡          (          ω                      t                    )                    +      ω              (                              cos            ⁡            (            ω                          t                        )            +            1                                              cos                              2                                      ⁡            (            ω                          t                        )                          )                            (        cos        ⁡                  (          ω                      t                    )                +        1        )                    2              =            ω      (      2      cos      ⁡      (      ω              t            )      −              cos                  3                    ⁡      (      ω              t            )      +      1      )                                cos                      2                          ⁡        (        ω                  t                )            (      cos      ⁡      (      ω              t            )      +      1              )                  2                      .      (  1  )If we will multiply that we must get on the expression             (      1      +      cos      ⁡      (      ω              t            )              (      1      +      cos      ⁡      (      ω              t            )      , therefore, the result will be:                    ω                  (          −                      cos                          2                                ⁡                      (            ω            t            )                    +          cos          ⁡                      (            ω            t            )                    +          1          )                            1        +        cos        ⁡                  (          ω          t          )                      =            ω              (        −                  cos                      2                          ⁡                  (          ω          t          )                +        cos        ⁡                  (          ω          t          )                +        1        −                  cos                      3                          ⁡        (        ω                  t                )        +                  cos                      2                          ⁡        (        ω                  t                )        +        cos        ⁡        (        ω                  t                )        )                    (      1      +      cos      ⁡      (      ω              t            )              )                  2                      =    =            ω              (        1        −                  cos                      3                          ⁡        (        ω                  t                )        +        2        cos        ⁡        (        ω                  t                )        )                    (      1      +      cos      ⁡      (      ω              t            )              )                  2                      .      (  2  )As we can see in (2) the multiplier             cos              2              ⁡    (    ω          t        )   is absent, therefore, the misprint takes a place.

How can I simply this : (omega sin(omega t)tan(omega t)+omega(cos(omega t)+1) sec^2(omega t))/((cos(omega t)+1)^2)

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