$\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are 3 unit vectors such that each is inclined at an angle $\theta$ with the other. A unit vector $\overrightarrow{d}$is equally inclined with these vectors at an angle $\alpha ,\text{ then }4\cos \theta -3\cos 2\alpha \text{ is }$

$\text{ If }\overrightarrow{d}=\alpha \overrightarrow{a}+\beta \overrightarrow{b}+\gamma \overrightarrow{c}\text{, by symmetry }\alpha =\beta =\gamma =k$

$\therefore \overrightarrow{d}=k(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})\Rightarrow \overrightarrow{d}\cdot \overrightarrow{a}=k(1+\overrightarrow{b}\cdot \overrightarrow{a}+\overrightarrow{c}\cdot \overrightarrow{a})$

$\cos \alpha =k(1+2\cos \theta )\dots \dots \left(1\right)$

$\overrightarrow{d}\cdot \overrightarrow{d}=3K\cos \alpha \Rightarrow 3K\cos \alpha =1\cdots \dots \dots \text{ (2) }$

$\text{ from (1) and (2) }3{\cos }^2\alpha =1+2\cos \theta$

$\Rightarrow 3\left(\frac{1+\cos 2\alpha }{2}\right)=1+2\cos \theta$

$\Rightarrow 3+3\cos 2\alpha =2+4\cos \theta \Rightarrow 4\cos \theta -3\cos 2\alpha =1$