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))

Payton George 2022-10-21 Answered
Simplifying an expression: Stuck with it
I 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 .
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Answers (1)

lipovicai1w
Answered 2022-10-22 Author has 9 answers
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've used the fact that ω r = 1 L C = 1 1 1 = 1.
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