Question

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

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

Answer 1

Answer: (A), (B)

$\frac{sin^4 x}{2}+\frac{cos^4}{3}=\frac{1}{5}$
$\Rightarrow \frac{sin^4 x}{2}+\frac{(1-sin^2 x)^2}{3}=\frac{1}{5}$
$\Rightarrow \frac{sin^4}{2}+\frac{1+sin^4 x-2sin^2 x}{3}=\frac{1}{5}$
$\Rightarrow 5 sin^4 x -4sin^2 x+2=\frac{6}{5}$
$\Rightarrow 25 sin^4 x-20sin^2 x+4=0$
$\Rightarrow (5 sin^2 x-2)^2$=0
$\Rightarrow sin^2 x=\frac{2}{5}$
$ sin^2 x=\frac{3}{5}, tan^2 x=\frac{2}{3}$
$\frac{sin^8 x}{8}+\frac{cos^8 x}{27}=\frac{1}{125}$