If cos–1$\alpha$ + cos–1$\beta$ + cos–1γ = 3$\pi$, then $\alpha$($\beta$ + γ) + $\beta$(γ + $\alpha$) + γ($\alpha$ + $\beta$) equals ______.

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If cos–1$\alpha$ + cos–1$\beta$ + cos–1γ = 3$\pi$, then $\alpha$($\beta$ + γ) + $\beta$(γ + $\alpha$) + γ($\alpha$ + $\beta$) equals 6. Explanation: We have cos–1$\alpha$ + cos–1$\beta$ + cos–1γ = 3$\pi$ $\Rightarrow$ cos–1$\alpha$ + cos–1$\beta$ + cos–1γ = $\pi$ + $\pi$ + $\pi$ $\Rightarrow$ cos–1$\alpha$ = $\pi$, cos–1$\beta$ = $\pi$ and cos–1γ = $\pi$ $\Rightarrow$ $\alpha$ = cos $\pi$, $\beta$ = cos $\pi$ and γ = cos $\pi$ $\therefore$ $\alpha$ = – 1, $\beta$ = – 1 and γ = – 1 Which gives $\alpha$ = $\beta$ = γ = –1 So $\alpha$($\beta$ + γ) + $\beta$(γ + $\alpha$) + γ($\alpha$ + $\beta$) $\Rightarrow$ (– 1)(– 1 – 1) + (– 1)(– 1 – 1) + (– 1)(– 1 – 1) $\Rightarrow$ (– 1)(– 2) + (– 1)(– 2) + (– 1)(– 2) $\Rightarrow$ 2 + 2 + 2 $\Rightarrow$ 6

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