TITLE: Trigonometric Equation - can this be solved using complex numbers
QUESTION [0 upvotes]: How do I solve the following:
$\cos (12x) = 5 \sin (3x) + 9 \tan^2( x )+ \cot ^2 (x)$ for $x \in (0,360)$
I tried converting cos and sin term into single angle i.e. into x but the equation becomes messy. Any elegant solution at hand?

REPLY [2 votes]: Not much different form Toby Mak's answer but  We could infact rewrite the given equaltion in the following form  $$1-\cos 12x+5(\sin 3x+1)+{\left( 3\tan x-\cot x\right)}^2=0$$ as each terms is $\ge 0$ we must have that each term must be zero ,
can you take it from here?

REPLY [0 votes]: $$a+b\geq 2\sqrt{ab}\\9tan^2x+1cot^2x\geq2\sqrt{9tan^2x\times 1cot^2x}=6$$so
$$cos(12x)-5sin(3x)\geq 6 $$Implicit $$cos(12x)-5sin(3x)\leq \max\{cos(12x)\}+\max \{ -5sin(3x)\}\leq 1+5$$ and only possibilities : $$\begin{cases}cos(12x)=1\\sin(3x)=-1\end{cases}$$
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