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

Problem

sin4(x)−cos4(x)+cos2(x)

Solution

1. Factor the first two terms as a difference of squares using the identity a2−b2=(a−b)*(a+b)

sin4(x)−cos4(x)=(sin2(x)−cos2(x))*(sin2(x)+cos2(x))

2. Apply the Pythagorean identity sin2(x)+cos2(x)=1 to simplify the expression.

(sin2(x)−cos2(x))*(1)+cos2(x)

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

sin2(x)−cos2(x)+cos2(x)

4. Cancel the opposite terms −cos2(x) and cos2(x)

sin2(x)

Final Answer

sin4(x)−cos4(x)+cos2(x)=sin2(x)

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