Question

if sin theta + sin ^2 theta =1 then prove that cos ^12 theta +3 cos^10 theta + 3 cos ^8 theta +cos ^6 theta -1 =0​

Answer 1

sin theta = 1- sin ^2 theta = cos^2 theta

(cos⁴ theta)³ + 3cos⁴ theta. cos²theta(cos⁴ theta+ cos ² theta) + (cos ² theta) ³-1

{ as (a+b) ³= a³+3ab+b³}

(cos⁴ theta+cos² theta) ³-1

((cos² theta) ²+cos ² theta) ³-1

(sin² theta +cos ² theta) ³-1

{ as sin ² theta+ cos ² theta= 1}

1-1=0= RHS

hence proved

hope this helps!