Certainly! Let's solve the equation step by step:
Given:
[tex]\[
\sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} + \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} = 2 \sec \theta
\][/tex]
1. Simplifying the expression:
Observe both terms inside the square roots:
[tex]\[
\sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} \quad \text{and} \quad \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}}
\][/tex]
2. Multiplication of terms:
Let's consider:
[tex]\[
x = \sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}}
\][/tex]
Then,
[tex]\[
\frac{1}{x} = \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}}
\][/tex]
The given equation becomes:
[tex]\[
x + \frac{1}{x} = 2 \sec \theta
\][/tex]
3. Multiplying the equation by [tex]\( x \)[/tex]:
To get rid of the fraction, multiply both sides by [tex]\( x \)[/tex]:
[tex]\[
x^2 + 1 = 2 x \sec \theta
\][/tex]
4. Using [tex]\( x \sec \theta \)[/tex]:
To isolate the trigonometric function, move the terms:
[tex]\[
x^2 + 1 - 2 x \sec \theta = 0
\][/tex]
5. Observing [tex]\( x \sec \theta \)[/tex]:
Try to solve this equation for [tex]\( x \)[/tex] which involves the secant function. But remember that [tex]\( \sec \theta = \frac{1}{\cos \theta} \)[/tex], we might seek an identity, considering known values of [tex]\( \sin \theta \)[/tex] and [tex]\( \cos \theta \)[/tex].
6. Solving for particular values:
There could be restrictions or specific interval checks of [tex]\( \theta \)[/tex] to verify if it leads to valid results.
However, upon deeper examination of the equation itself and solving within the method constraints, it turns out that there are no solutions for [tex]\( \theta \)[/tex] that satisfy this equation.
Therefore, the equation:
[tex]\[
\sqrt{\frac{1 + \sin \theta}{1 - \sin \theta}} + \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} = 2 \sec \theta
\][/tex]
has no solutions.
So, there are no [tex]\( \theta \)[/tex] values for which the equation holds true.
[tex]\[
\boxed{[]}
\][/tex]
