To solve the equation [tex]\(\sin \theta + \frac{\cos^2 \theta}{1 + \sin \theta} = 1\)[/tex] for [tex]\(\theta\)[/tex], let's go through the following steps:
1. Simplify the Equation:
Start with the given equation:
[tex]\[
\sin \theta + \frac{\cos^2 \theta}{1 + \sin \theta} = 1
\][/tex]
2. Combine the Fractions:
Combine the terms on the left-hand side under a common denominator:
[tex]\[
\frac{(1 + \sin \theta) \sin \theta + \cos^2 \theta}{1 + \sin \theta} = 1
\][/tex]
We know that [tex]\(\cos^2 \theta = 1 - \sin^2 \theta\)[/tex]. Substitute this identity into the equation:
[tex]\[
\frac{(1 + \sin \theta) \sin \theta + 1 - \sin^2 \theta}{1 + \sin \theta} = 1
\][/tex]
3. Simplify the Numerator:
Expand and simplify the numerator:
[tex]\[
(1 + \sin \theta) \sin \theta + 1 - \sin^2 \theta = \sin \theta + \sin^2 \theta + 1 - \sin^2 \theta
\][/tex]
[tex]\[
\sin \theta + 1
\][/tex]
So the equation becomes:
[tex]\[
\frac{\sin \theta + 1}{1 + \sin \theta} = 1
\][/tex]
4. Compare Numerator and Denominator:
Since the numerator [tex]\((\sin \theta + 1)\)[/tex] and the denominator [tex]\((1 + \sin \theta)\)[/tex] are identical, the fraction simplifies to 1:
[tex]\[
1 = 1
\][/tex]
This simplification suggests that the equation is always true regardless of the value of [tex]\(\theta\)[/tex].
5. Interpreting the Solution:
Since [tex]\(1 = 1\)[/tex] is an identity equation, it indicates that any value of [tex]\(\theta\)[/tex] should satisfy the given equation.
6. Checking for Extraneous Solutions:
However, the result of analyses indicates that there might be constraints or specific intervals within the domain of [tex]\(\theta\)[/tex] where the equation holds but doesn't necessarily suggest valid or specific solutions within standard ranges and settings of [tex]\(\theta\)[/tex].
Hence the equation [tex]\(\sin \theta + \frac{\cos^2 \theta}{1 + \sin \theta} = 1\)[/tex] has solutions, but within typical constraints or practical intervals such used in trigonometric identities, we have not found specific traditional angle values universally valid for all contexts. Therefore, it implies no real specific angles were solved from this general identity validation.
So, the set of solutions [tex]\( \theta\)[/tex] in standard range returns:
[tex]\[
\boxed{\text{No specific values found within constraints}}
\][/tex]
