To solve the equation [tex]\( 2 \sin^2 \left( \frac{\pi}{4} - \frac{\theta}{2} \right) = 1 - \sin \theta \)[/tex], we will use trigonometric identities and algebraic manipulation. Let's go through the solution step-by-step:
1. Recall the double-angle identity for sine:
[tex]\[\sin^2 x = \frac{1 - \cos(2x)}{2}\][/tex]
2. Set [tex]\( x = \frac{\pi}{4} - \frac{\theta}{2} \)[/tex] and note that the left side of the equation uses [tex]\( \sin^2 \left( \frac{\pi}{4} - \frac{\theta}{2} \right) \)[/tex]:
[tex]\[
2 \sin^2 \left( \frac{\pi}{4} - \frac{\theta}{2} \right)
\][/tex]
3. Using the sine square identity:
[tex]\[
\sin^2 \left( \frac{\pi}{4} - \frac{\theta}{2} \right) = \frac{1 - \cos \left( 2\left(\frac{\pi}{4} - \frac{\theta}{2}\right) \right)}{2}
\][/tex]
4. Simplify the argument of the cosine:
[tex]\[
2 \left( \frac{\pi}{4} - \frac{\theta}{2} \right) = \frac{\pi}{2} - \theta
\][/tex]
5. Thus:
[tex]\[
\sin^2 \left( \frac{\pi}{4} - \frac{\theta}{2} \right) = \frac{1 - \cos (\frac{\pi}{2} - \theta)}{2}
\][/tex]
6. Recall that [tex]\( \cos (\frac{\pi}{2} - \theta) = \sin \theta \)[/tex], hence:
[tex]\[
\sin^2 \left( \frac{\pi}{4} - \frac{\theta}{2} \right) = \frac{1 - \sin \theta}{2}
\][/tex]
7. Substitute back into the original equation:
[tex]\[
2 \sin^2 \left( \frac{\pi}{4} - \frac{\theta}{2} \right) = 1 - \sin \theta
\][/tex]
[tex]\[
2 \left( \frac{1 - \sin \theta}{2} \right) = 1 - \sin \theta
\][/tex]
8. Simplify inside the parentheses:
[tex]\[
1 - \sin \theta = 1 - \sin \theta
\][/tex]
This shows the equation holds for all values of [tex]\(\theta\)[/tex].
9. Conclusion:
The given equation:
[tex]\[
2 \sin^2 \left( \frac{\pi}{4} - \frac{\theta}{2} \right) = 1 - \sin \theta
\][/tex]
is an identity that holds true for all [tex]\(\theta\)[/tex]. Therefore, there are no specific solutions to this equation; it is true universally for any [tex]\(\theta\)[/tex].
