Question

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

• A. ${\tan }^{2}x=\frac{3}{2}$
• B. $\frac{{\sin }^{8}x}{8}+\frac{{\cos }^{8}x}{27}=\frac{1}{125}$
• C. ${\tan }^{2}x=\frac{2}{3}$
• D. $\frac{{\sin }^{8}x}{8}+\frac{{\cos }^{8}x}{27}=\frac{2}{27}$

Answer 1

Answer: (B), (C)

The correct options are
B $\frac{{\sin }^{8}x}{8}+\frac{{\cos }^{8}x}{27}=\frac{1}{125}$
C ${\tan }^{2}x=\frac{2}{3}$

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

$\Rightarrow 15{\sin }^{4}x+10(1-{\sin }^{2}x{)}^2=6$ [$\because$ multiply both side by $5$]

$\Rightarrow 25{\sin }^{4}x-20{\sin }^{2}x+4=0\Rightarrow (5{\sin }^{2}x-2{)}^2=0$

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

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

${\tan }^{2}x=\frac{2}{3}$