Simplify

Problem

(sin3(θ)+cos3(θ))/(sin(θ)+cos(θ))

Solution

1. Identify the numerator as a sum of cubes in the form a3+b3 where a=sin(θ) and b=cos(θ)

2. Apply the formula for the sum of cubes: a3+b3=(a+b)*(a2−a*b+b2)

3. Substitute the trigonometric functions into the formula to rewrite the numerator:

sin3(θ)+cos3(θ)=(sin(θ)+cos(θ))*(sin2(θ)−sin(θ)*cos(θ)+cos2(θ))

4. Cancel the common factor (sin(θ)+cos(θ)) from the numerator and the denominator:

((sin(θ)+cos(θ))*(sin2(θ)−sin(θ)*cos(θ)+cos2(θ)))/(sin(θ)+cos(θ))=sin2(θ)−sin(θ)*cos(θ)+cos2(θ)

5. Apply the identity sin2(θ)+cos2(θ)=1 to simplify the remaining expression.

6. Simplify the final expression by replacing the sum of the squared terms with 1

Final Answer

(sin3(θ)+cos3(θ))/(sin(θ)+cos(θ))=1−sin(θ)*cos(θ)

Want more problems? Check here!