# Inequality method to solve trigonometric equation.

Ok, while seeing examples in my book concerning with the general solution of trigonometric equations, I saw they used this inequality method . What is this??? The book didn't give any precise elaboration of this method,however it solved many equations with this method. To clarify,some examples:$$(\tan x)^4 + (\tan y)^4 + 2(\cot x\cot y)^2 = 3 + (\sin(x+y))^2;$$ the book solved this using $$(\tan x)^4 +(\tan y)^4 +2(\cot x\cot y)^2 \geq 4 \text{ and } 3 + {\sin(x+y)}^2 \leq 4$$ in order to have the equality $$(\tan x)^2 = (\tan y)^2 = 1,$$ thus $$x = y = n\pi + \frac{\pi}{4}.$$ Another illustration $$2(\cos x\sin 2x)^2 = x^2 + x^{-2}$$ where
$0<x \leq \frac{\pi}{2}$. The inequality method goes like $$2(\cos x\sin 2x)^2 < 2$$ and $$x^2 + x^{-2} \geq 2,$$ so they have no solution.

Still another one, $(\sin x)^6 = 1 + (\cos 3x)^4$, again using inequality method $$(\sin x)^6 \leq 1 \text{ and } 1 + (\cos 3x)^4 \geq 1 , x = (2n + 1)\frac{\pi}{2}.$$ And so many.

But,nothing did the book write regarding this method. When I came before $$\sin 7x = \sin 3x + \sin x \text{, x is in the close interval of 0 and \pi},$$ I wanted to use this method,but soon was at the blues as I could not find any way to use this inequality method . My question is what is this inequality method all about? Can I use it in any equation to solve it? If not so,when can I use it? Plz help explaining me what it is & when it can be used .

• Use backslash to correctly write the trigonometric functions: instead of $\;sin x\;$ , write a backslash immediately before the "s" and get $\;\sin x$ – Timbuc Oct 9 '14 at 17:29

The general idea is that you want to solve the equality $f(x)=g(x)$. Then you find such a number $a$ so $\forall x f(x)\le a$ and $\forall x g(x)\ge a$. So, in order to guarantee that the equation holds you need to have $x$ such that $f(x)=g(x)=a$. Now you can solve this equality for $x$. Note that the system $$\begin{cases}f(x)=a,\\g(x)=a,\end{cases}$$ is not always compatible.
• I doubt that your last problem can be solved with method. Just try to build the plots for left hand side and right hand side (wolframalpha.com). Anyway, lefthand side is in $[-1,1]$ and the right hand side is non-negative, so such an $a$ as described in my answer does not exist. – TZakrevskiy Oct 10 '14 at 12:30