Verify the Identity cos(x)^4-sin(x)^4=1-2sin(x)^2

Problem

cos(x)−sin(x)=1−2*sin(x)

Solution

1. Recognize the difference of squares on the left side of the equation, where the expression is in the form a2−b2 with a=cos(x) and b=sin(x)

cos(x)−sin(x)=(cos(x)−sin(x))*(cos(x)+sin(x))

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

(cos(x)−sin(x))*(1)=cos(x)−sin(x)

3. Substitute the identity cos(x)=1−sin(x) into the expression to express everything in terms of sin(x)

(1−sin(x))−sin(x)

4. Combine like terms to reach the final form of the identity.

1−2*sin(x)

Final Answer

cos(x)−sin(x)=1−2*sin(x)

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