P and q are positive real numbers.

Let p = (sinx) 2 p = (cosx) 2 x ≠ 0 and x≠kπ/2(k is an integer).

√( 1+p^2)+√( 1+q^2)

= 1/|sinx|+ 1/|cosx|

& gt=2√[ 1/(|sinx|*|cosx|)]

| sinx | * | cosx | & lt=[(sinx)^2+(cosx)^2]/2= 1/2

so 2√[ 1/(| sinx | * | cosx |)]& gt； =2√2

That is, √ (1+p2)+√ (1+Q2) > = 2 √ 2.