To solve the equation [tex]\(\csc^4 \theta - \csc^2 \theta = \cot^4 \theta + \cot^2 \theta\)[/tex], we will proceed step-by-step using trigonometric identities.
Firstly, we know the following trigonometric identities:
1. [tex]\(\csc^2 \theta = 1 + \cot^2 \theta\)[/tex] which implies [tex]\(\cot^2 \theta = \csc^2 \theta - 1\)[/tex].
Given the equation:
[tex]\[
\csc^4 \theta - \csc^2 \theta = \cot^4 \theta + \cot^2 \theta
\][/tex]
Let [tex]\(x = \csc^2 \theta\)[/tex]. This means:
[tex]\[
\csc^4 \theta = x^2
\][/tex]
Also, since [tex]\(\cot^2 \theta = \csc^2 \theta - 1\)[/tex], we have:
[tex]\[
\cot^4 \theta = (\csc^2 \theta - 1)^2 = (x - 1)^2
\][/tex]
Let's substitute these into the given equation:
[tex]\[
x^2 - x = (x - 1)^2 + (x - 1)
\][/tex]
Expanding the right-hand side:
[tex]\[
(x - 1)^2 + (x - 1) = (x^2 - 2x + 1) + (x - 1)
\][/tex]
[tex]\[
= x^2 - 2x + 1 + x - 1
\][/tex]
[tex]\[
= x^2 - x
\][/tex]
So the equation simplifies to:
[tex]\[
x^2 - x = x^2 - x
\][/tex]
It turns out that we have an identity:
[tex]\[
x^2 - x = x^2 - x
\][/tex]
This equation is true for any value of [tex]\(x\)[/tex]. Therefore, [tex]\(\csc^2 \theta\)[/tex] can take any real value satisfying the constraint [tex]\( \csc^2 \theta \geq 1 \)[/tex] because cosecant of an angle is always greater than or equal to 1 or less than or equal to -1 for real [tex]\(\theta\)[/tex].
Let's summarize our findings:
[tex]\[
\csc^2 \theta \geq 1 \implies \cot^2 \theta = \csc^2 \theta - 1 \geq 0
\][/tex]
For [tex]\(\csc \theta = \pm 1\)[/tex] or higher real values satisfying the identity for [tex]\(\csc \theta \neq 0\)[/tex], the form [tex]\( \cot^2 \theta\)[/tex] and [tex]\( \csc^2 \theta \)[/tex] uphold the identity.
Therefore, the given trigonometric equation is satisfied for all values of [tex]\(\theta\)[/tex] where [tex]\(\operatorname{cosec} \theta\)[/tex] and hence [tex]\(\cot \theta\)[/tex] maintain the necessary conditions of being real.
