Question

Solve the following equations.
$2si{n}^{2}2x+6si{n}^{2}x=5.$

Answer 1

$2si{n}^{2}2x+6si{n}^{2}x=5$

$\Rightarrow 8si{n}^{2}x.co{s}^{2}x+6si{n}^{2}x=5$

$\Rightarrow ut$ $t=si{n}^{2}x$

$\Rightarrow 8t(1-t)+6t=5$

$\Rightarrow 8t-8t^2+6t=5$

$\Rightarrow 8t^2-14t+5=0$

$\Rightarrow 8t^2-10t-4t+5=0$

$\Rightarrow 2t(4t-5)-1(4t-5)=0$

$\Rightarrow (2t-1)(4t-5)=0$

$t=1/2t=5/4$

$si{n}^{2}x=1/2$

$\Rightarrow sinx=\pm 1/\sqrt{2}$

$\Rightarrow x=x\pi +(-1{)}^n\pm \pi /4$

$=x\pi \pm (-1{)}^n\frac{\pi }{4}$

$t=5/4$

$\Rightarrow sinx=\pm \frac{\sqrt{5}}{2}>1$ hence no solution.

$\therefore$ only solution is $x=x\pi \pm (-1{)}^n\frac{\pi }{4};xϵz$