We aim to simplify the expression:

$\sin^4(x) - \cos^4(x) + \cos^2(x).$

Step 1: Apply the difference of squares

The term $\sin^4(x) - \cos^4(x)$ can be factored using the difference of squares formula: $a^2 - b^2 = (a - b)(a + b).$

Let $a = \sin^2(x)$ and $b = \cos^2(x)$, so: $\sin^4(x) - \cos^4(x) = (\sin^2(x) - \cos^2(x))(\sin^2(x) + \cos^2(x)).$

Step 2: Simplify using trigonometric identities

From the Pythagorean identity: $\sin^2(x) + \cos^2(x) = 1.$

Substituting this into the factorization: $\sin^4(x) - \cos^4(x) = (\sin^2(x) - \cos^2(x))(1).$

Thus: $\sin^4(x) - \cos^4(x) = \sin^2(x) - \cos^2(x).$

Step 3: Combine the remaining terms

The expression now becomes: $\sin^2(x) - \cos^2(x) + \cos^2(x).$

Combine like terms: $\sin^2(x) - \cos^2(x) + \cos^2(x) = \sin^2(x).$

Final Answer:

$\sin^4(x) - \cos^4(x) + \cos^2(x) = \sin^2(x).$

Would you like me to elaborate on any step?

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Related Questions:

1. What is the difference of squares formula, and how is it used in trigonometry?
2. How does the Pythagorean identity simplify trigonometric expressions?
3. Can $\sin^2(x) - \cos^2(x)$ be rewritten in terms of other trigonometric functions?
4. What are other ways to simplify $\sin^4(x) - \cos^4(x)$ without factoring?
5. How does this simplification apply to solving trigonometric equations?

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Tip:

Always check if a trigonometric expression can be simplified using identities like the Pythagorean identity or factorization techniques. These shortcuts save time and effort!