Question

If $y=\frac{1}{{\sin }^{6}x+{\cos }^{6}x},$ then $y$ can be

• A. $\frac{3}{2}$
• B. $2$
• C. $4$
• D. $5$

Answer 1

Answer: (A), (B), (C)

$4$

$y=\frac{1}{{\sin }^{6}x+{\cos }^{6}x}$
$=\frac{1}{({\sin }^{2}x+{\cos }^{2}x{)}^3-3{\sin }^{2}x{\cos }^{2}x({\sin }^{2}x+{\cos }^{2}x)}$
$=\frac{1}{1-3{\sin }^{2}x{\cos }^{2}x}$
$=\frac{1}{1-\frac{3(\sin 2x{)}^2}{4}}$
$\Rightarrow y=\frac{4}{4-3{\sin }^{2}2x}$

Since, $0\le {\sin }^{2}2x\le 1$
$\Rightarrow 1\le 4-3{\sin }^{2}2x\le 4\Rightarrow \frac{1}{4}\le \frac{1}{4-3{\sin }^{2}2x}\le 1$
$\Rightarrow y\in [1,4]$