Consider the system of equations
$$x \cos^{3} y+3x \cos y \sin^{2} y =14 $$
$$ x \sin^{3} y+3x \cos^{2} y \sin y = 13 $$
$1)$ the values of $x$ is /are..
Answer is $ \pm\sqrt 5 $
The number of values of $y$ in $(0,6 \pi)$is
Answer is $6$
$Sin^2 y + 2cos^2 y $ is
Answer is $ \frac95 $
I added both the equation and make whole cube
I subtracted both the equation and make whole cube then I divided these equations to find the value of $ \tan y $ = $ \dfrac{1}{2} $
I divided the first equation with $ cos^2 y $ and plugged the value of $\tan y$ which gives $x= +5\sqrt 5 $. But I also need –$ 5\sqrt 5 $ I have no idea where I go wrong.
Best Answer
I think your approach is quite good.
\begin{align*} \frac{x(\sin^3y+3\cos^2y\sin y)}{x(\cos^3y+3\cos y\sin^2y)}&=\frac{13}{14}\\ \frac{\tan^3y+3\tan y}{1+3\tan^2y}&=\frac{13}{14}\\ 14\tan^3y-39\tan^2y+42\tan y-13&=0\\ (2\tan y-1)(7\tan^2y-16\tan y+13)&=0 \end{align*}
$\tan y=\frac12$.
$y$ has $6$ values in $(0,6\pi)$.
$\sec^2y=1+(\frac12)^2=\frac{5}{4}$.
$\cos y=\pm\sqrt{\frac45}=\pm\frac{2}{\sqrt5}$.
$\sin y=\tan y\cos y=\pm\frac{1}{\sqrt5}$.
So, we have
\begin{align*} \pm x\left[\left(\frac{1}{\sqrt5}\right)^3+3\left(\frac{2}{\sqrt5}\right)^2\left(\frac{1}{\sqrt5}\right)\right]&=13\\ x&=\pm5\sqrt5 \end{align*}
$\sin^2y+2\cos^2y=(\frac{1}{\sqrt5})^2+2(\frac{2}{\sqrt5})^2=\frac95$.