A portfolio consists of three stocks: $X, Y, Z$. $40\%$ of the portfolio is invested in $X$, $40\%$ in $Y$, and $20\%$ in $Z$. The three stocks are uncorrelated. The volatilities of the three stocks are $0.27, 0.23, 0.44$, respectively. Calculate the correlation of $X$ with the portfolio.
In the provided solution, this is said:
The covariance of $X$ with the portfolio, since $X$ is uncorrelated with the other two stocks, is $0.4(0.27^2) = 0.02916$.
What justifies this statement? As far as I can tell this does not follow from $\mathrm{Cov}(A,B) = \mathrm{E}[AB] - \mathrm{E}[A]\mathrm{E}[B]$, and I don't know what other tools could be used here.
The covariance of $X$ with the portfolio is
$$E(X(0.4X+0.4Y+0.2Z))-E(X)E(0.4X+0.4Y+0.2Z)$$
Expand this and use the assumptions.