Question

if sinA+ sin^2A =1 then ahow that cos ^12 A+3cos^10A+3cos^8A+cos^6A=1

Answer 1

sinA+sin^2A=1
sinA=1-sin^2A=cos^2A
now,

LHS=cos^12A+3cos^10A+3cos^8A+cos^6A
={cos^2A}^6+3(cos^2A)(cos^2A)^4+3(cos^2A)^2(cosA)+cos^6A

=sin^6A+3sin^4A.cos^2A+3cos^4A.sinA+cos^6A

=(sin^2A+cos^2A)^3

because we know
sin^2A+cos^2A=1
us e this above

=(sin^2A+cos^2A)^3=(1)^3=1=RHS