Simplify sin(x)^4-cos(x)^4+cos(x)^2

Problem

sin^4(x)−cos^4(x)+cos^2(x)

Solution

1. Factor the first two terms as a difference of squares using the identity a^2−b^2=(a−b)*(a+b)

sin^4(x)−cos^4(x)=(sin^2(x)−cos^2(x))*(sin^2(x)+cos^2(x))

2. Apply the Pythagorean identity sin^2(x)+cos^2(x)=1 to simplify the expression.

(sin^2(x)−cos^2(x))*(1)+cos^2(x)

3. Simplify the resulting expression by removing the parentheses and combining like terms.

sin^2(x)−cos^2(x)+cos^2(x)

4. Cancel the opposite terms −cos^2(x) and cos^2(x)

sin^2(x)

Final Answer

sin^4(x)−cos^4(x)+cos^2(x)=sin^2(x)

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