Consider an equation with $x$ as variable such that $7\sin 3x - 2 \sin 9 x = \sec^2 \theta + 4\csc^2 \theta $. If the value of $\dfrac{15}{\pi}\text{(Minimum positive real x - Maximum negative real x)} = k$, then find the value of $\dfrac k 2$.
Attempt:
I have simplified the equation to $\sin 3x(1+ 8 \sin^23x)$ but I am really not able to understand what the question wants. Is it that we have to find the range of $x$. Then how do we do that?
I think I have it.
Take your simplified version letting $y=\sin3x$.
$y(1+8y^2)=\sec^2\theta+4\csc^2\theta$
Take the implicit derivative with respect to theta letting $y'=0$.
You get an equation for $tan^2\theta$. Each term on the right can be expressed in terms of $tan^2\theta$. Swap in the value from the previous step, you have an integer, q.
Now you have $y(1+8y^2)=q$ for some integer q, found from earlier steps.
q has a value allowing the use of The Rational Root Theorem to find appropriate values for y. Those values can be used to get x by solving $\sin3x=y_0$ where $y_0$ is one of the solutions from earlier.
There are an infinite number of $x$ values that will give you a solution to that last equation. One is a lowest maximum, another is a highest minimum. Take those values and plug tem into the above to get k, then divide by 2.
I think you'll get $k/2=5$.