To prove that [tex]\(\sin^4 \theta + 2\sin \theta \cos^2 \theta + \cos^4 \theta = 1\)[/tex], let's start by examining and simplifying the expression step by step.
Given expression:
[tex]\[ \sin^4 \theta + 2\sin \theta \cos^2 \theta + \cos^4 \theta \][/tex]
First, let's rewrite the given expression using common trigonometric identities. We know that:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
and we also know the power reduction formulas:
[tex]\[ \sin^4 \theta = (\sin^2 \theta)^2 \][/tex]
[tex]\[ \cos^4 \theta = (\cos^2 \theta)^2 \][/tex]
Let's simplify the middle term, [tex]\(2\sin \theta \cos^2 \theta\)[/tex], in the given expression to see if it can be rewritten in a simpler form.
Now, try substituting the trigonometric identity [tex]\(\cos^2 \theta = 1 - \sin^2 \theta\)[/tex] and see if we can simplify the expression.
Combining these parts:
[tex]\[ \sin^4 \theta + 2\sin \theta \cos^2 \theta + \cos^4 \theta \][/tex]
Rewrite [tex]\(\cos^2 \theta\)[/tex] in terms of [tex]\(\sin^2 \theta\)[/tex]:
[tex]\[ \cos^2 \theta = 1 - \sin^2 \theta \][/tex]
So, [tex]\(2\sin \theta \cos^2 \theta\)[/tex] becomes [tex]\(2\sin \theta (1 - \sin^2 \theta)\)[/tex]:
[tex]\[ = \sin^4 \theta + 2\sin \theta (1 - \sin^2 \theta) + (1 - \sin^2 \theta)^2 \][/tex]
Expand [tex]\(2\sin \theta (1 - \sin^2 \theta)\)[/tex]:
[tex]\[ = \sin^4 \theta + 2\sin \theta - 2\sin^3 \theta + (1 - 2\sin^2 \theta + \sin^4 \theta) \][/tex]
Combine like terms:
[tex]\[ = 2\sin^4 \theta - 2\sin^3 \theta - 2\sin^2 \theta + 2\sin \theta + 1 \][/tex]
We see that the terms do not easily all combine to 1. This suggests that the original equation might have been more misleading. Therefore, calculating [tex]\( \sin^4 \theta + 2\sin \theta \cos^2 \theta + \cos^4 \theta \)[/tex], we would conclude that it does not simplify directly to 1.
Therefore, after careful verification and recalculating, we conclude:
[tex]\[
\sin^4 \theta + 2\sin \theta \cos^2 \theta + \cos^4 \theta \neq 1
\][/tex]
Thus, the given statement [tex]\(\sin^4 \theta + 2\sin \theta \cos^2 \theta + \cos^4 \theta = 1\)[/tex] is not true.
