Question

The number of distinct solutions of the equation $\frac{5}{4}co{s}^{2}2x+co{s}^{4}x+si{n}^{4}x+co{s}^{6}x+si{n}^{6}x=2$ in the interval $[0,2\pi ]$ is___

Answer 1

$\frac{5}{4}co{s}^{2}2x+co{s}^{4}x+si{n}^{4}x+co{s}^{6}x+si{n}^{6}x=2$
$\Rightarrow \frac{5}{4}co{s}^{2}2x+(si{n}^{2}x+co{s}^{2}x{)}^2-2si{n}^{2}xco{s}^{2}x$
$+(si{n}^{2}x+co{s}^{2}x{)}^3-3si{n}^{2}xco{s}^{2}x(si{n}^{2}x+co{s}^{2}x)=2\Rightarrow \frac{5}{4}co{s}^{2}x+1-\frac{1}{2}si{n}^{2}2x+1-\frac{3}{4}si{n}^{2}2x=2$
$\Rightarrow co{s}^{2}2x=si{n}^{2}2x$
$\Rightarrow ta{n}^{2}2x=1$
$\Rightarrow tan2x=\pm 1$
$\Rightarrow 2x=n\pi \pm \frac{\pi }{4},n\in Z$
$\Rightarrow x=(4n\pm 1)\frac{\pi }{8},n\in z$
$\Rightarrow x=\frac{\pi }{8},\frac{3\pi }{8},\frac{5\pi }{8},\frac{7\pi }{8},\frac{9\pi }{8},\frac{11\pi }{8},\frac{13\pi }{8},\frac{15\pi }{8}$