How to solve for the value of $x$ for the following equation $$\frac {(\sec x + \tan x)^2 – (\sec2x + \tan x)^2}{\sin2x – \sin x} = 2$$
My Attempt: $$ \frac {\sec^{2}x + 2\sec x\tan x + \tan^{2}x – (\frac {1 + \tan^{2}x}{1 – \tan^{2}x} + \tan x)^2}{\frac {2\tan x}{1 + \tan^{2}x} – \sin x} = 2$$
$$ \implies \frac {\sec^{2}x + 2\sec x\tan x + \tan^{2}x – (\frac {1 + \tan^{2}x}{1 – \tan^{2}x})^2 – 2\tan x(\frac {1 + \tan^{2}x}{1 – \tan^{2}x}) – \tan^{2}x}{\frac {2\tan x – \sin x(1 + \tan^{2}x)}{1 + \tan^{2}x}} = 2 $$
But here I stopped. I couldn't find any easier way for simplifying it more as it was growing more complicated while I was trying to solve. My other friends also tried to solve but their final equation turned into complex one.
Clarification: I thought I would lead to a conclusion with bringing an individual trig function. My intention was to solve it via using '$\tan x$' but couldn't transform '$\sec x$' & '$\sin x$' into $'\tan x'$ using any trig identities and formula.
Assumption: There is perhaps a basic in starting point I am missing for holding. Multiplying or dividing with something could have been lot more easier to end with a good conclusion.
Or if that not so, can we move straightforward from the last part of my approach to get the result?
Best Answer
Not a pleasant solution but a nasty problem.
Consider that you look for the zero's of function $$f(x)=(\sec (x) + \tan (x))^2 - (\sec (2x) + \tan (x))^2-2(\sin (2x) - \sin(x)) $$ If you try that tangent half-angle substitution, you will end with a polynomial of degree $14$ $$5 t^{14}-3 t^{13}-55 t^{12}-2 t^{11}+193 t^{10}+19 t^9-307 t^8+36 t^7+367 t^6+19 t^5-85 t^4-2 t^3+11 t^2-3 t-1=0$$ which does not factor; so, forget such an approach.
A problem is the discontinuities because of the secants and the tangents. To remove all of them, multiply everything by $\big[\cos^2(x)\, \cos^2(2x)\big]$. Using the multiple angle formula, we end with $$g(x)=9 \sin (x)-2 \sin (2 x)-3 \sin (3 x)-6 \sin (4 x)+5 \sin (5 x)-2 \sin (6x)+$$ $$\sin (7 x)-\sin (8 x)-4 \cos (2 x)+4 \cos (4 x)=0$$ which looks mor pleasant.
In the interval $-\pi \leq x \leq \pi$, there are three obvious solutions $x=-\pi$, $x=0$ and $x=\pi$. Plotting $g(x)$ we can now notice "close" to $x=-0.42$, to $x=0.91$ and to $x=2.15$.
At this point, to polish the roots, I don not see anything elese than Newton method (or any other root-finding procedure). For each root, Newton iterates will be $$\{-0.41,-0.417499,-0.417459\}$$ $$\{0.91,0.907893,0.907886\}$$ $$\{2.15,2.135306,2.135701\}$$