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simplify this $r = \frac{C_p}{C_v} = \frac{C_v + R}{C_v} = \frac{R}{r} + 1 + \frac{R}{R/r} + 1$

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Find trigonometric ratios using a Pythagorean or reciprocal identity

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Q. simplify this

r = \frac{C_p}{C_v} = \frac{C_v + R}{C_v} = \frac{R}{r} + 1 + \frac{R}{R/r} + 1

Simplify Compound Expression: We are given the expression $r = \frac{C_p}{C_v}$ and are asked to simplify the compound expression $\frac{C_p}{C_v} = \frac{C_v + R}{C_v} = \frac{R}{r} + 1 + \frac{R}{R/r} + 1$ . Let's start by simplifying the right-hand side of the equation step by step.
Combine Terms: First, we simplify the term $Cv + \frac{R}{Cv}$ . Since $Cv$ is a common factor, we can combine the terms as follows: $\newline$ $Cv + \frac{R}{Cv} = Cv(1 + \frac{R}{Cv^2})$
Substitute for $r$ : Next, we simplify the term $\frac{R}{r} + 1$ . Since $r = \frac{C_p}{C_v}$ , we can substitute $\frac{C_p}{C_v}$ for $r$ : $\frac{R}{r} + 1 = \frac{R}{(\frac{C_p}{C_v})} + 1$
Correct Previous Mistake: Now, we simplify the term $R/R/r + 1$ . Since $r = C_p/C_v$ , we can substitute $C_p/C_v$ for $r$ : $\newline$ $\frac{R}{R}/\frac{C_p}{C_v} + 1$
Final Simplification: We notice that there is a mistake in the previous step. The term $R/R/r$ is not correctly simplified. We should have simplified it as follows: $\newline$ $R/R/r + 1 = 1/r + 1$ $\newline$ Since $r = C_p/C_v$ , we substitute $C_p/C_v$ for $r$ : $\newline$ $1/r + 1 = 1/(C_p/C_v) + 1$

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