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Note that $\beta$ is the coefficient of the portfolio regressed on the benchmark. That is \begin{align*} r_P = \alpha+\beta r_B + \varepsilon, \end{align*} where $\varepsilon$ is the residual. Then standard deviation of the residual is called the residual risk. Specifically, \begin{align*} std(\varepsilon) &= \sqrt{var(r_P-\beta r_B)}\\ &=\sqrt{\sigma_P^2 + \beta^2 \sigma_B^2 - 2 \beta \rho(r_P, r_B) \sigma_P\sigma_B}\\ &=\sqrt{\sigma_P^2 + \beta^2 \sigma_B^2 - 2 \beta\, \beta\, var(r_B)}\\ &= \sqrt{\sigma_P^2 - \beta^2 \sigma_B^2}, \end{align*} since $cov(r_P, r_B) = \rho(r_P, r_B)\sigma_P \sigma_B =\beta\, var(r_B)$.

Note that $\beta$ is the coefficient of the portfolio regressed on the benchmark. That is \begin{align*} r_P = \alpha+\beta r_B + \varepsilon, \end{align*} where $\varepsilon$ is the residual. The standard deviation of the residual is called the residual risk. Specifically, \begin{align*} std(\varepsilon) &= \sqrt{var(r_P-\beta r_B-\alpha)}\\ &=\sqrt{\sigma_P^2 + \beta^2 \sigma_B^2 - 2 \beta \rho(r_P, r_B) \sigma_P\sigma_B}\\ &=\sqrt{\sigma_P^2 + \beta^2 \sigma_B^2 - 2 \beta\, \beta\, var(r_B)}\\ &= \sqrt{\sigma_P^2 - \beta^2 \sigma_B^2}, \end{align*} since $cov(r_P, r_B) = \rho(r_P, r_B)\sigma_P \sigma_B =\beta\, var(r_B)$.

Note that $\beta$ is the coefficient of the portfolio regressed on the benchmark. That is \begin{align*} r_P = \beta r_B + \varepsilon, \end{align*} where $\varepsilon$ is the residual. Then standard deviation of the residual is called the residual risk. Specifically, \begin{align*} std(\varepsilon) &= \sqrt{var(r_P-\beta r_B)}\\ &=\sqrt{\sigma_P^2 + \beta^2 \sigma_B^2 - 2 \beta \rho(r_P, r_B) \sigma_P\sigma_B}\\ &=\sqrt{\sigma_P^2 + \beta^2 \sigma_B^2 - 2 \beta\, \beta\, var(r_B)}\\ &= \sqrt{\sigma_P^2 - \beta^2 \sigma_B^2}, \end{align*} since $cov(r_P, r_B) = \rho(r_P, r_B)\sigma_P \sigma_B =\beta\, var(r_B)$.

Note that $\beta$ is the coefficient of the portfolio regressed on the benchmark. That is \begin{align*} r_P = \alpha+\beta r_B + \varepsilon, \end{align*} where $\varepsilon$ is the residual. Then standard deviation of the residual is called the residual risk. Specifically, \begin{align*} std(\varepsilon) &= \sqrt{var(r_P-\beta r_B)}\\ &=\sqrt{\sigma_P^2 + \beta^2 \sigma_B^2 - 2 \beta \rho(r_P, r_B) \sigma_P\sigma_B}\\ &=\sqrt{\sigma_P^2 + \beta^2 \sigma_B^2 - 2 \beta\, \beta\, var(r_B)}\\ &= \sqrt{\sigma_P^2 - \beta^2 \sigma_B^2}, \end{align*} since $cov(r_P, r_B) = \rho(r_P, r_B)\sigma_P \sigma_B =\beta\, var(r_B)$.

Note that $\beta$ is the coefficient of the portfolio regressed on the benchmark. That is \begin{align*} r_P = \beta r_B + \varepsilon, \end{align*} where $\varepsilon$ is the residual. Then standard deviation of the residual is called the residual risk. Specifically, \begin{align*} std(\varepsilon) &= \sqrt{var(r_P-\beta r_B)}\\ &=\sqrt{\sigma_P^2 + \beta^2 \sigma_B^2 - 2 \beta \rho(r_P, r_B) \sigma_P\sigma_B}\\ &=\sqrt{\sigma_P^2 + \beta^2 \sigma_B^2 - 2 \beta\, \beta\, var(r_B)}\\ &= \sqrt{\sigma_P^2 - \beta^2 \sigma_B^2}, \end{align*} since $cov(r_P, r_B) = \rho(r_P, r_B)\sigma_B \sigma_B =\beta\, var(r_B)$.

Note that $\beta$ is the coefficient of the portfolio regressed on the benchmark. That is \begin{align*} r_P = \beta r_B + \varepsilon, \end{align*} where $\varepsilon$ is the residual. Then standard deviation of the residual is called the residual risk. Specifically, \begin{align*} std(\varepsilon) &= \sqrt{var(r_P-\beta r_B)}\\ &=\sqrt{\sigma_P^2 + \beta^2 \sigma_B^2 - 2 \beta \rho(r_P, r_B) \sigma_P\sigma_B}\\ &=\sqrt{\sigma_P^2 + \beta^2 \sigma_B^2 - 2 \beta\, \beta\, var(r_B)}\\ &= \sqrt{\sigma_P^2 - \beta^2 \sigma_B^2}, \end{align*} since $cov(r_P, r_B) = \rho(r_P, r_B)\sigma_P \sigma_B =\beta\, var(r_B)$.