Simplifying an expression: Stuck with itI have to prove that the expression ω C − 1 ω L ω C − 1 ω L + ω L − 1 ω C is equal to 1 3 − ( ( ω r ω ) 2 + ( ω ω r ) 2 ) where ω r = 1 L C .

Question

Simplifying an expression: Stuck with itI have to prove that the expression            ω      C      −              1                  ω          L                            ω      C      −              1                  ω          L                    +      ω      L      −              1                  ω          C                    is equal to      1          3      −      (      (                        ω          r                ω                    )        2            +      (              ω                  ω          r                            )        2            )      where       ω    r    =      1          L      C      .

lipovicai1w · Accepted Answer

The expressions are NOT equal when   ω  =  2 and   L  =  1 and   C  =  1:            ω      C      −              1                  ω          L                            ω      C      −              1                  ω          L                    +      ω      L      −              1                  ω          C                          =                2      ⋅      1            −                    1                  2          ⋅          1                            2      ⋅      1            −                    1                  2          ⋅          1                          +            2      ⋅      1            −                    1                  2          ⋅          1                          =                3      2                    4      −      1                  =          1    2  and      1          3            −                    [                              (                                          ω                r                            ω                        )                    2                +                              (                          ω                              ω                r                                      )                    2                ]                =        1          3            −                    [                              (                          1              2                        )                    2                +                              (                          2              1                        )                    2                ]                =        1                3            −                    1        4                  −            4                =        1                −              5        4                      =    −      4    5    ,where I&#039;ve used the fact that         ω    r    =      1          L      C        =      1          1      ⋅      1              =  1.

Simplifying an expression: Stuck with it I have to prove that the expression (omega C - 1/(omega L))/(omega C - 1/(omega L) + omega L - 1(omega C)) is equal to 1/(3-( ((omega_r)/(omega))^2 + ((omega)/(omega_r))^2)) where omega_r= 1/(sqrt(LC))

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