Question

If $\frac{si{n}^{4}x}{2}+\frac{co{s}^{4}x}{3}=\frac{1}{5}$, then

• A. $si{n}^{2}x=\frac{2}{5}$
• B. $co{s}^{2}x=\frac{3}{5}$
• C. $\frac{si{n}^{6}x}{4}+\frac{co{s}^{6}x}{4}=1$
• D. None

Answer 1

Answer: (A), (B)

The correct options are
A

$si{n}^{2}x=\frac{2}{5}$

B

$co{s}^{2}x=\frac{3}{5}$

Multiplying the given relation by 4
$\frac{4si{n}^{4}x}{2}+\frac{4co{s}^{4}x}{3}=\frac{4}{5}\Rightarrow \frac{(1-cos2x{)}^2}{2}+\frac{(1+cos2x{)}^2}{3}=\frac{4}{5}\Rightarrow cos2x=\frac{1}{5}\therefore 2co{s}^{2}x=\frac{6}{5}and2si{n}^{2}x=\frac{4}{5}$
$co{s}^{2}x=\frac{3}{5}$ and $si{n}^{2}x=\frac{2}{5}$