Question

Solve the following equations.
Solve the equation $4cosx(2-3si{n}^{2}x)+(cos2x+1)=0.$ Find the least distance between its positive roots.

Answer 1

$4cosx(2-3si{n}^{2}x)+cos2x+1=0$

$4cosx(2-3(1-co{s}^{2}x\left)\right)+2co{s}^{2}x=0$

$4cosx(3co{s}^{2}x-1)+2co{s}^{2}x=0$

$\Rightarrow 12co{s}^{3}x-4cosx+2co{s}^{2}x=0$

$\Rightarrow 2cosx(6co{s}^{2}x+cosx-2)=0$

$2cosx=0$

$6co{s}^{2}x+cosx-2=0$

$cox=0$

$cox=\frac{-1\pm \sqrt{1+48}}{12}$

$x=(2x+1)\pi /2$ $=\frac{-1\pm 7}{12}$

$=\pi /2,3\pi /2...=1/2--2/3$

$x=2x\pi \pm \pi /3,2x\pi \pm co{s}^{-1}(-2/3)$

$=\pi /3,....$

Least distance b/w +ve roots $=\pi /2-\pi /3$

$=\pi /6$