<s>[INST] Is $\csc x = a+bi$ defined? A question in my assignment asks to find which equation can have its roots  $\sec^2 x, \csc^2 x$ (no restrictions in $x$ were given).
The equations were quadratic ($x^2+bx+c$) and the one which satisfies $-b=c$ can be the required equations.
There were two which satisfies this,
$x^2-3x+3=0, x^2-9x+9=0$
The second one is absolutely correct but the first in the first one $\csc^2x$ is in the form of $a+bi$. And so first one wasn't the correct answer.
But I am having doubt in the answer, as by Euler's formula,
$$\sin x=\frac{e^{ix}-e^{-ix}}{2i}$$
And here putting a suitable complex value of $x$, $\sin x$ can have a value in form of $a+bi$ and so $\csc^2 x$.
So, am I correct? Can we put a complex value in trigonometric functions or can they have a complex value? And also tell what if $\sin x=2$?
 [/INST] If you want to solve the equation $$\csc (x+i y)=a+i\,b$$ expand first the cosecant, go to hyperbolic functions, use the conjugate, separate the real and imaginary parts to end with
$$\frac{\sin (x) \cosh (y)}{\cos ^2(x) \sinh ^2(y)+\sin
   ^2(x) \cosh ^2(y)}=a \tag 1$$
$$\frac{\cos (x) \sinh (y)}{\cos ^2(x) \sinh ^2(y)+\sin
   ^2(x) \cosh ^2(y)}=-b \tag 2$$ Computing the ratio
$$\tan (x) \coth (y)=-\frac ab\quad \implies \quad \color{blue}{y=-\coth ^{-1}\left(\frac{a }{b}\cot (x)\right)} \tag 3$$ Replacing $y$ in $(1)$ give the simple
$$\csc (x) \sqrt{1-\frac{b^2 }{a^2}\tan
   ^2(x)}=\frac{a^2+b^2}{a}$$ Squaring
$$\csc ^2(x)-\frac{b^2 }{a^2}\sec ^2(x)=\left(\frac{a^2+b^2}{a}\right)^2$$ Cross multiply by $\sin^2(x)\,\cos^2(x)$
$$\cos ^2(x)-\frac{b^2 }{a^2}\sin ^2(x)=\left(\frac{a^2+b^2}{a}\right)^2\sin^2(x)\,\cos^2(x)$$ Let $x=\sin^{-1}(t)$ to obtain
$$t^4-\frac{a^2+b^2+1}{a^2+b^2} t^2+\frac{a^2}{\left(a^2+b^2\right)^2}=0$$ which is just a quadratic in $t^2$.
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