Question

Solve the following equations.
Find (in degrees) the solution x of the equation $1+cos10xcos6x=2co{s}^{2}8x+si{n}^{2}8x,$

Answer 1

$1+cos10xcos6x=2co{s}^{2}8x+si{n}^{2}8x$

$1+cos10xcos6x=co{s}^{2}8x+1$

$\Rightarrow cos10xcos6x=co{s}^{8x}$

$\Rightarrow \frac{1}{2}(cos16x+cos4x)=co{s}^{2}8x$

$\Rightarrow cos16x+cos4x=2co{s}^{2}8x$

$u=4x$

$\Rightarrow cos4u+cosu=2cos2u$

$\Rightarrow 2co{s}^{2}u-1+cosu=2co{s}^{2}2u$

$\Rightarrow cosu=1$

$\Rightarrow u=2x\pi \Rightarrow 4x=2x\pi$

$\Rightarrow x=4\pi /2$

$x=\frac{x\pi }{2}$ & $xϵ({20}^{\circ },{80}^{\circ })$

$x=0\Rightarrow x=0$

$x=1\Rightarrow x=\pi /2={90}^{\circ }$

$\Rightarrow$ No solution in $({20}^{\circ },{80}^{\circ })$