Trigonometric equation and how to solve them in generalI have to prove the equality sin ⁡ 4 x 1 + cos ⁡ 4 x . cos ⁡ 2 x 1 + cos ⁡ 2 x . cos ⁡ x 1 + cos ⁡ x = tan ⁡ x 2 I looked at the fractions individually and got ( sin ⁡ 4 x + tan ⁡ 4 x ) . ( cos ⁡ 2 x + 1 ) . ( cos ⁡ x + 1 ) = tan ⁡ x 2 which I can't simplify further.I looking for a solution to this problem and if someone can refer me to a document where it is explained how to go about solving different kinds of trigonometric equations.

Question

Trigonometric equation and how to solve them in generalI have to prove the equality             sin      ⁡      4      x              1      +      cos      ⁡      4      x        .            cos      ⁡      2      x              1      +      cos      ⁡      2      x        .            cos      ⁡      x              1      +      cos      ⁡      x        =  tan  ⁡      x    2  I looked at the fractions individually and got   (      sin    ⁡    4    x    +      tan    ⁡    4    x    )  .  (      cos    ⁡    2    x    +    1    )  .  (  cos  ⁡  x  +  1  )  =  tan  ⁡      x    2   which I can&#039;t simplify further.I looking for a solution to this problem and if someone can refer me to a document where it is explained how to go about solving different kinds of trigonometric equations.

kavdawg8w8 · Accepted Answer

Prove the equality                    (1)                                          sin                        4            x                                1            +            cos                        4            x                          ⋅                              cos                        2            x                                1            +            cos                        2            x                          ⋅                              cos                        x                                1            +            cos                        x                                              =        tan        ⁡                              x            2                              Using just one known identity,                                          sin                        x                                1            +            cos                        x                                              =        tan        ⁡                              x            2                              together with the definition   tan    x  =            sin            x              cos            x      , LHS of       (1)   can be transformed as                                  =        tan        ⁡                                            4              x                        2                          ⋅                              cos                        2            x                                1            +            cos                        2            x                          ⋅                              cos                        x                                1            +            cos                        x                                                            =                              sin                        2            x                                cos                        2            x                          ⋅                              cos                        2            x                                1            +            cos                        2            x                          ⋅                              cos                        x                                1            +            cos                        x                                                            =                              sin                        2            x                                1            +            cos                        2            x                          ⋅                              cos                        x                                1            +            cos                        x                                                            =        tan        ⁡                              2            x                    2                ⋅                              cos                        x                                1            +            cos                        x                                                            =                              sin                        x                                cos                        x                          ⋅                              cos                        x                                1            +            cos                        x                                                            =                              sin                        x                                1            +            cos                        x                          =        …

ritualizi6zk · Accepted Answer

Notation:   sin  ⁡  (  n  x  )  =      s    n   and similar for cos, and tan. Also let   t  =  tan  ⁡  (  x      /    2  )  =      t          1              /            2      . Then                    (1)                              s          1                =                              2            t                                1            +                          t              2                                      ,                  c          1                =                              1            −                          t              2                                            1            +                          t              2                                          Then each factor in your fractions can be reduced to being a function of just       c    1    ,      s    1  , e.g:      s    4    =  2      s    2        c    2    =  4      s    1        c    1    (  2      c    1    2    −  1  )  1  +      c    4    =  2      c    2    2    =  2  (  1  −      s    2    2    )  =  2  (  1  −  4      s    1    2        c    1    2    )Using the above equations   (  1  ) should get you to the answer.

Trigonometric equation and how to solve them in general I have to prove the equality (sin 4x)/(1+cos 4x)(cos 2x)/(1+cos 2x)(cos x)/(1+cos x)=tan (x/2)

Answered question

Answer & Explanation

New Questions in Trigonometry