What is sin 4θ - cos 4θ equal to for any real number θ?
1 - 2 cos 2θ
Simplifying \(\sin^4\theta - \cos^4\theta\): A Trigonometry Solution
Let's simplify the given expression \(\sin^4\theta - \cos^4\theta\). This expression involves powers of sine and cosine functions.
The expression can be rewritten as \((\sin^2\theta)^2 - (\cos^2\theta)^2\). This looks like a difference of squares, which follows the algebraic identity: \(a^2 - b^2 = (a-b)(a+b)\).
Applying this identity, where \(a = \sin^2\theta\) and \(b = \cos^2\theta\), we get:
\( \sin^4\theta - \cos^4\theta = (\sin^2\theta - \cos^2\theta)(\sin^2\theta + \cos^2\theta) \)
Now, we use a fundamental trigonometric identity that relates \(\sin^2\theta\) and \(\cos^2\theta\):
- The Pythagorean identity states that for any real number \(\theta\), \(\sin^2\theta + \cos^2\theta = 1\).
Substitute this identity into our simplified expression:
\( (\sin^2\theta - \cos^2\theta)(\sin^2\theta + \cos^2\theta) = (\sin^2\theta - \cos^2\theta)(1) \)
So, the expression simplifies to \(\sin^2\theta - \cos^2\theta\).
We need to express this result in the form of the given options. The options are generally in terms of either \(\sin^2\theta\) only or \(\cos^2\theta\) only, or a combination that simplifies differently.
Let's try to express \(\sin^2\theta - \cos^2\theta\) solely in terms of \(\cos^2\theta\). We can use the identity \(\sin^2\theta = 1 - \cos^2\theta\).
Substitute \(\sin^2\theta = 1 - \cos^2\theta\) into \(\sin^2\theta - \cos^2\theta\):
\( \sin^2\theta - \cos^2\theta = (1 - \cos^2\theta) - \cos^2\theta \)
\( = 1 - \cos^2\theta - \cos^2\theta \)
\( = 1 - 2\cos^2\theta \)
Alternatively, let's try to express \(\sin^2\theta - \cos^2\theta\) solely in terms of \(\sin^2\theta\). We can use the identity \(\cos^2\theta = 1 - \sin^2\theta\).
Substitute \(\cos^2\theta = 1 - \sin^2\theta\) into \(\sin^2\theta - \cos^2\theta\):
\( \sin^2\theta - \cos^2\theta = \sin^2\theta - (1 - \sin^2\theta) \)
\( = \sin^2\theta - 1 + \sin^2\theta \)
\( = 2\sin^2\theta - 1 \)
Now let's compare our derived forms (\(1 - 2\cos^2\theta\) and \(2\sin^2\theta - 1\)) with the given options:
- Option 1: \(1\) (Does not match)
- Option 2: \(1 - 2 \sin^2\theta\) (Does not match \(2\sin^2\theta - 1\))
- Option 3: \(2 \cos^2\theta + 1\) (Does not match \(1 - 2\cos^2\theta\))
- Option 4: \(1 - 2 \cos^2\theta\) (Matches our derived form)
Therefore, \(\sin^4\theta - \cos^4\theta\) is equal to \(1 - 2\cos^2\theta\) for any real number \(\theta\). Note that \(2\sin^2\theta - 1\) is also a valid simplification, but it is not provided as an option.
Revision Table: Key Trigonometric Identities
| Identity Name | Formula | Use Case |
|---|---|---|
| Pythagorean Identity | \(\sin^2\theta + \cos^2\theta = 1\) | Relating \(\sin^2\theta\) and \(\cos^2\theta\), simplifying expressions |
| Difference of Squares | \(a^2 - b^2 = (a-b)(a+b)\) | Factoring expressions |
| Rearranged Pythagorean Identity | \(\sin^2\theta = 1 - \cos^2\theta\) | Expressing \(\sin^2\theta\) in terms of \(\cos^2\theta\) |
| Rearranged Pythagorean Identity | \(\cos^2\theta = 1 - \sin^2\theta\) | Expressing \(\cos^2\theta\) in terms of \(\sin^2\theta\) |
Additional Information: Understanding Trigonometric Simplification
Simplifying trigonometric expressions often involves using fundamental identities. The goal is usually to reduce the expression to a simpler form or to express it in terms of specific trigonometric functions or powers.
Key strategies include:
- Recognizing algebraic patterns (like difference of squares, perfect squares, factoring).
- Applying fundamental identities (Pythagorean identity, reciprocal identities, quotient identities).
- Converting all terms to sine and cosine if necessary.
- Using identities to change the form (e.g., expressing everything in terms of \(\sin\theta\) or \(\cos\theta\)).
In this problem, recognizing \(\sin^4\theta - \cos^4\theta\) as a difference of squares was the crucial first step. This allowed us to use the powerful Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\) to simplify one factor, leaving only \(\sin^2\theta - \cos^2\theta\). The final step was using another form of the Pythagorean identity to express this result in terms of only \(\cos^2\theta\) to match the given options.
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