If cos θ 1 + cos θ 2 + cos θ 3 + cos θ 4 = - 4 , then the value of cot θ 1 2 + cot…

Given,

$\cos \left({\theta }_1\right)+ \cos \left({\theta }_2\right)+\cos \left({\theta }_3\right)+ \cos \left({\theta }_4\right)=-4$

We know that,

$1\le \cos \left(\theta \right)\le 1$

$\cos \left({\theta }_1\right)=-1,\cos \left({\theta }_2\right)=-1,\cos \left({\theta }_3\right)=-1,\cos \left({\theta }_4\right)=-1$

${\theta }_1=\mathrm{\pi }= {\theta }_2= {\theta }_3= {\theta }_4$

$\Rightarrow \cot \left(\frac{\mathrm{\pi }}{2}\right)+\cot \left(\frac{\mathrm{\pi }}{2}\right)+\cot \left(\frac{\mathrm{\pi }}{2}\right)+\cot \left(\frac{\mathrm{\pi }}{2}\right)\Rightarrow 4\cot \left(\frac{\mathrm{\pi }}{2}\right)$

$\Rightarrow 0 \left(\because \cot \left(\frac{\mathrm{\pi }}{2}\right)=0\right)$