If cosx + cos²x = 1,the numerical value of (sin¹² + 3sin¹⁰x + 3sin⁸x + sin⁶x - 1) = ?

Solution (By Examveda Team)

$$\eqalign{ & {\sin ^{12}} + 3{\sin ^{10}}x + 3{\sin ^8}x + {\sin ^6}x - 1 \cr & \Rightarrow {\left( {{{\sin }^4}x + {{\sin }^2}x} \right)^3} - 1 \cr & \Rightarrow {\left( {{{\cos }^2}x + {{\sin }^2}x} \right)^3} - 1 \cr & \left[ {\cos x + {{\cos }^2}x = 1} \right] \cr & (\cos x = 1 - {\cos ^2}x = {\sin ^2}x) \cr & \Rightarrow 1 - 1 \cr & \Rightarrow 0 \cr} $$