If the triangle PQR varies, then the minimum value of cos(P+Q)+cos(Q+R)+cos(R+P) is

The correct option is A󔼨Given PQR is a triangle.Hence, P+Q+R=π∴P+Q=π−R, Q+R=π−P, P+R=π−Q∴E=cos(P+Q)+cos(Q+R)+cos(P+R)=cos(π−R)+cos(π−P)+cos(π−Q)=−cosR−cosP−cosQ=−(cosP+cosQ+cosR)To minimize the given expression (which is under negative sign), we need to maximize cosP+cosQ+cosR.We know that cosP+cosQ+cosR will be maximum when cosP=cosQ=cosR⇒P=Q=R=π3 (i.e, PQR is an equilateral triangle)∴Emin=−cos(π3)=−12Hence, option A is correct.