Question

If sin x + sin² x = 1, then write the value of cos⁸ x + 2 cos⁶ x + cos⁴ x.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ \sin x+{\sin }^{2}x=1\left(1\right) \\ \Rightarrow \sin x=1-{\sin }^{2}x \\ \Rightarrow \sin x=co{s}^{2}x\left(2\right) \\ \mathrm{Now},\mathrm{taking}\mathrm{square}\mathrm{of}\left(1\right): \\ \Rightarrow {\left(\sin x+{\sin }^{2}x\right)}^2={\left(1\right)}^2 \\ \Rightarrow {\left(\sin x\right)}^2+{\left({\sin }^{2}x\right)}^2+2\left(\sin x\right)\left({\sin }^{2}x\right)=1 \\ \Rightarrow {\left(\sin x\right)}^2+{\left(\sin x\right)}^4+2{\left(\sin x\right)}^3=1 \\ \Rightarrow {\left(\sin x\right)}^2+2{\left(\sin x\right)}^3+{\left(\sin x\right)}^4=1 \\ \Rightarrow {\left({\text{cos}}^{2}x\right)}^2+2{\left({\text{cos}}^{2}x\right)}^3+{\left({\text{cos}}^{2}x\right)}^4=1 \\ \Rightarrow {\text{cos}}^{4}x+2{\text{cos}}^{6}x+{\text{cos}}^{8}x=1 \\ \therefore {\text{cos}}^{8}x+2{\text{cos}}^{6}x+{\text{cos}}^{4}x=1 \end{gathered}$$