Given #tanP/tanQ=1/5=>tanQ=5tanP#
Now #LHS=cot(P+Q)=1/tan(P+Q)#
#=(1-tanPtanQ)/(tanP+tanQ)#
#=(1-5tanPxxtanP)/(tanP+5tanP)#
#=(1-5tan^2P)/(6tanP)#
#=((1-5tan^2P)xxcos^2P)/(6tanPxxcos^2P)#
#=(cos^2P-5sin^2P)/(6sinPxxcosP)#
#=(6cos^2P-5cos^2P-5sin^2P)/(3xxsinPxxcosP)#
#=(6cos^2P-5(cos^2P+sin^2P))/(3xxsin2P)#
#=1/3csc(2P)(6cos^2P-5)=RHS#
Proved