Let's solve the given trigonometric equation step-by-step:
[tex]\[
\tan^2(\theta) - \tan^2(\theta) \sin^2(\theta) = \sin^2(\theta)
\][/tex]
To simplify this equation, we can use the known trigonometric identity:
[tex]\[
\tan^2(\theta) = \frac{\sin^2(\theta)}{1 - \sin^2(\theta)}
\][/tex]
Let [tex]\( x = \sin^2(\theta) \)[/tex]. Then, we have:
[tex]\[
\tan^2(\theta) = \frac{x}{1 - x}
\][/tex]
We substitute this into the original equation:
[tex]\[
\left(\frac{x}{1 - x}\right) - \left(\frac{x}{1 - x} \cdot x\right) = x
\][/tex]
Now we need to simplify the left-hand side of the equation:
[tex]\[
\left(\frac{x}{1 - x}\right) - \left(\frac{x^2}{1 - x}\right)
\][/tex]
Since both terms in the left-hand side have a common denominator, we can combine them:
[tex]\[
\frac{x - x^2}{1 - x}
\][/tex]
Now, simplifying the fraction:
[tex]\[
\frac{x(1 - x)}{1 - x}
\][/tex]
Provided that [tex]\(1 - x \neq 0\)[/tex], we can cancel out [tex]\(1 - x\)[/tex] in the numerator and denominator:
[tex]\[
x = x
\][/tex]
The simplified equation [tex]\( x = x \)[/tex] holds true for any [tex]\( x \)[/tex], which in this context means any [tex]\(\sin^2(\theta)\)[/tex].
Therefore, the original equation:
[tex]\[
\tan^2(\theta) - \tan^2(\theta) \sin^2(\theta) = \sin^2(\theta)
\][/tex]
holds true for any [tex]\( \theta \)[/tex] where [tex]\( \sin^2(\theta) \)[/tex] is defined.
