What is the maximum value of sin⁴ θ + cos⁴ θ ?

1

⇒ sin⁴ θ + cos⁴ θ

= ( sin² θ + cos² θ)² - 2sin² θ cos² θ

= 1 - 2sin² θ cos² θ

= 1 - \(\frac{1}{2}\) ( 2sin θ cos θ )²

= 1 - \(\frac{1}{2}\) ( sin 2θ )²

For maximum value of expression, \(\frac{1}{2}\) ( sin2 θ )² should be minimum

i.e. only possible, when sin2 θ = 0

∴ maximum value is 1.