Find the greatest value of sin⁴θ + cos⁴θ.

Question

Accepted Answer

We want to find the maximum value of sin⁴θ + cos⁴θ.

Step 1: Use the identity a⁴ + b⁴ = (a²)² + (b²)².
Let x = sin²θ and y = cos²θ.
Then sin⁴θ + cos⁴θ = x² + y².

Step 2: Since sin²θ + cos²θ = 1, we have x + y = 1.

Step 3: We want to maximize x² + y² given x + y = 1.

Step 4: Using the identity:
x² + y² = (x + y)² - 2xy = 1² - 2xy = 1 - 2xy.

Step 5: To maximize x² + y², minimize xy.

Step 6: Since x and y are sin²θ and cos²θ, both between 0 and 1, and x + y = 1, the minimum product xy occurs when one of them is 0 and the other is 1.

Step 7: So minimum xy = 0.

Step 8: Therefore, maximum value of sin⁴θ + cos⁴θ = 1 - 2*0 = 1.

Hence, the greatest value of sin⁴θ + cos⁴θ is 1.

Answer

Answer

Answer