Certainly! Let's prove that the given trigonometric expression simplifies to 1:
[tex]\[ 2 \sin ^2 \theta + \cos ^2 2 \theta + 2 \sin ^2 \theta \cdot \cos 2 \theta = 1 \][/tex]
### Step-by-Step Solution:
1. Rewrite the given expression:
[tex]\[ 2 \sin ^2 \theta + \cos ^2 2 \theta + 2 \sin ^2 \theta \cdot \cos 2 \theta \][/tex]
2. Recall known trigonometric identities:
- [tex]\(\cos 2 \theta = 1 - 2 \sin^2 \theta\)[/tex]
- [tex]\(\cos^2 x + \sin^2 x = 1\)[/tex]
- [tex]\(\cos^2 2\theta = (1 - 2\sin^2 \theta)^2\)[/tex]
3. Calculate [tex]\(\cos^2 2\theta\)[/tex] using the identity for [tex]\(\cos 2\theta\)[/tex]:
[tex]\[ \cos 2 \theta = 1 - 2 \sin^2 \theta \][/tex]
[tex]\[ \cos^2 2 \theta = (1 - 2 \sin^2 \theta)^2 \][/tex]
[tex]\[ \cos^2 2 \theta = 1 - 4 \sin^2 \theta + 4 \sin^4 \theta \][/tex]
4. Substitute [tex]\(\cos^2 2\theta\)[/tex] and simplify the given expression:
[tex]\[ 2 \sin ^2 \theta + \cos ^2 2 \theta + 2 \sin ^2 \theta \cdot \cos 2 \theta \][/tex]
[tex]\[ = 2 \sin ^2 \theta + (1 - 4 \sin^2 \theta + 4 \sin^4 \theta) + 2 \sin ^2 \theta (1 - 2 \sin^2 \theta) \][/tex]
[tex]\[ = 2 \sin^2 \theta + (1 - 4 \sin^2 \theta + 4 \sin^4 \theta) + 2 \sin^2 \theta - 4 \sin^4 \theta \][/tex]
5. Combine like terms:
[tex]\[ = 2 \sin^2 \theta + 2 \sin^2 \theta + 1 - 4 \sin^2 \theta + 4 \sin^4 \theta - 4 \sin^4 \theta \][/tex]
[tex]\[ = (2 \sin^2 \theta + 2 \sin^2 \theta - 4 \sin^2 \theta) + (4 \sin^4 \theta - 4 \sin^4 \theta) + 1 \][/tex]
[tex]\[ = 0 + 0 + 1 \][/tex]
[tex]\[ = 1 \][/tex]
### Conclusion:
[tex]\[ 2 \sin ^2 \theta + \cos ^2 2 \theta + 2 \sin ^2 \theta \cdot \cos 2 \theta = 1 \][/tex]
We have thus proved that the given trigonometric expression simplifies to 1.
