The maximum and minimum value of sin⁴θ+cos⁴θ are...

Explanation:

Given the trigonometry expression sin⁴θ+cos⁴θ , the minimum value of the expression occurs at when θ = 0° and maximum value occurs at when θ = 90°.

From trigonometry identity;

sin²θ+cos²θ = 1

square both sides;

(sin²θ+cos²θ)² = 1²

open the parenthesis;

sin⁴θ+2sin²θcos²θ+cos⁴θ = 1

subtract 2sin²θcos²θ from both sides;

sin⁴θ+2sin²θcos²θ+cos⁴θ-2sin²θcos²θ = 1-2sin²θcos²θ

sin⁴θ+cos⁴θ = 1-2sin²θcos²θ

Since the minimum value occur at θ = 0°, substitute θ = 0° into the right hand side of the resulting expression as shown;

sin⁴θ+cos⁴θ = 1-2sin²(0)cos²(0)

sin⁴θ+cos⁴θ = 1-2(0)(1)

sin⁴θ+cos⁴θ = 1-0

sin⁴θ+cos⁴θ = 1

Hence the minimum value of sin⁴θ+cos⁴θ  is 1

For the maximum value;

Since the maximum value occur at θ = 90°, substitute θ = 90° into the right hand side of the resulting expression as shown;

sin⁴θ+cos⁴θ = 1-2sin²(90)cos²(90)

sin⁴θ+cos⁴θ = 1-2(1)(0)

sin⁴θ+cos⁴θ = 1-0

sin⁴θ+cos⁴θ = 1

Hence the maximum value of sin⁴θ+cos⁴θ  is also 1