Question

What is the minimum value of sin square theta + cos ^ 4 theta?​

Answer 1

sin²θ + cos⁴θ

⇒ sin²θ + (cos²θ)²

⇒ sin²θ + (1 - sin²θ)²

⇒ sin²θ + [1 + sin⁴θ - 2sin²θ]

Removing the brackets:

⇒ [sin²θ - 2sin²θ] + 1 + sin⁴θ

⇒  - sin²θ + 1 + sin⁴θ

Re-arranging the terms:

⇒ (1 - sin²θ) + sin⁴θ

⇒ cos²θ + sin⁴θ

Here, Trigonometric Identities are used:

1. sin²θ + cos²θ = 1 ⇒        cos²θ = (1 - sin²θ)
2. (a - b)² = a² + b² - 2ab