Simplify 1+(sin(x)-cos(x))(sin(x)+cos(x))

Problem

1+(sin(x)−cos(x))*(sin(x)+cos(x))

Solution

1. Identify the difference of squares pattern in the expression (sin(x)−cos(x))*(sin(x)+cos(x))

2. Apply the formula (a−b)*(a+b)=a2−b2 to the product.

(sin(x)−cos(x))*(sin(x)+cos(x))=sin2(x)−cos2(x)

3. Substitute this result back into the original expression.

1+sin2(x)−cos2(x)

4. Apply the Pythagorean identity 1−cos2(x)=sin2(x) to simplify the expression further.

sin2(x)+(1−cos2(x))

5. Combine the like terms.

sin2(x)+sin2(x)=2*sin2(x)

Final Answer

1+(sin(x)−cos(x))*(sin(x)+cos(x))=2*sin2(x)

Want more problems? Check here!