Question

$If\frac{{\sin }^{4}x}{2}+\frac{{\cos }^{4}x}{3}=\frac{1}{5},thenprovethat\frac{{\sin }^{8}x}{8}+\frac{{\cos }^{8}x}{27}=\frac{1}{125}$

Answer 1

$\frac{{\sin }^{4}x}{2}+\frac{{\cos }^{4}x}{3}=\frac{1}{5}$

Let ${\sin }^{2}x=a$

$\therefore {\cos }^{2}x=1-a$

$\frac{a^2}{2}+\frac{(1-a{)}^2}{3}=\frac{1}{5}$

$15a^2+10(1-a{)}^2=6$

$25a^2-20a+4=0$

$(5a-2{)}^2=0$

$\Rightarrow a=\frac{2}{5}$

$\therefore {\sin }^{2}x=\frac{2}{5}$ and ${\cos }^{2}x=\frac{3}{5}$

$\frac{{\sin }^{8}x}{8}+\frac{{\cos }^{8}x}{27}=\frac{16}{625}\cdot \frac{1}{8}+\frac{81}{625}\cdot \frac{1}{27}$

$=\frac{2}{625}+\frac{3}{625}=\frac{1}{625}$

$\therefore \frac{{\sin }^{8}x}{8}+\frac{{\cos }^{8}x}{27}=\frac{1}{125}$