Mathematics - Trigonometric Equation Question with Solution | TestHub

$$\begin{array}{rl}& \frac{{\sin }^{4}x}{2}+\frac{{\cos }^{4}x}{3}=\frac{1}{5}\\ & \Rightarrow \frac{{\sin }^{4}x}{2}+\frac{{\left(1-{\sin }^{2}x\right)}^2}{3}=\frac{1}{5}\\ & \Rightarrow \frac{{\sin }^{4}x}{2}+\frac{1+{\sin }^{4}x-2{\sin }^{2}x}{3}=\frac{1}{5}\\ & \Rightarrow 5{\sin }^{4}x-4{\sin }^{2}x+2=\frac{6}{5}\\ & \Rightarrow 25{\sin }^{4}x-20{\sin }^{2}x+4=0\\ & \Rightarrow {\left(5{\sin }^{2}x-2\right)}^2=0\\ & \Rightarrow {\sin }^{2}x=\frac{2}{5},{\cos }^{2}x=\frac{3}{5},{\tan }^{2}x=\frac{2}{3}\\ & \therefore \frac{{\sin }^{8}x}{8}+\frac{{\cos }^{8}x}{27}=\frac{1}{125}\\ & \end{array}$$