I have a equation which is defined as $\langle\langle x_ix_j\rangle^M\langle \cos(\theta)\rangle^C\rangle^M$ where $M$ is the McCormick envelope of product of variables $x_ix_j$ and $C$ is defined as envelope of cosine function.
The McCormick envelope is defined as \begin{align*} \langle x_ix_j\rangle^M &= \{ \alpha \ge x_i^lx_j+x_j^lx_i-x_i^lx_j^l \\ &\qquad \alpha \ge x_i^ux_j+x_j^ux_i-x_i^ux_j^u \\ &\qquad \alpha \le x_i^lx_j+x_j^ux_i-x_i^lx_j^u \\ &\qquad \alpha \le x_i^ux_j+x_j^lx_i-x_i^ux_j^l\}, \end{align*} where $x^l,x^u$ are constants for both $x_i,x_j$ and known.
The envelope for cosine function is defined as \begin{align*} \langle \cos(\theta)\rangle^C &= \{\beta \le 1-\frac{1-\cos(\theta^m)}{(\theta^m)^2}\theta^2\\ &\qquad \beta\ge\frac{\cos(\theta^l)-\cos(\theta^u)}{(\theta^l-\theta^u)}(\theta-\theta^l)+\cos(\theta^l)\}, \end{align*} where $\theta^l,\theta^u,\theta^m$ are constants and known.
Now, I have to define the McCormick envelope of $\langle\langle x_ix_j\rangle^M\langle cos(\theta)\rangle^C\rangle^M$. Is it equivalent/true to write $\langle\langle x_ix_j\rangle^M\langle \cos(\theta \rangle^C\rangle^M$ as \begin{align*} \langle\langle x_ix_j\rangle^M\langle \cos(\theta)\rangle^C\rangle^M = \langle \alpha \beta \rangle^M &= \{ \gamma \ge \alpha^l\beta+\beta^l\alpha-\alpha^l\beta^l \\ &\qquad \gamma \ge \alpha^u\beta+\beta^u\alpha-\alpha^u\beta^u \\ &\qquad \gamma \le \alpha^l\beta+\beta^u\alpha-\alpha^l\beta^u \\ &\qquad \gamma \le \alpha^u\beta+\beta^l\alpha-\alpha^u\beta^l\}? \end{align*} If this is true, then how to define $\alpha^l,\alpha^u,\beta^l,\beta^u$? Do the first two inequalities of $\langle x_ix_j\rangle^M$ becomes the lower limit for $\alpha$ and the last two ineqaulites of $\langle x_ix_j\rangle^M$ becomes upper limit of $\alpha$? Or is there any other/better way to define McCormick envelope for such equation?