The maximum value of sin2θ + cos2θ is

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Expression
= sin⁴θ + cos⁴θ
= (sin²θ)² + (cos²θ)²
= (sin²θ + cos²θ)² – 2 sin²θ.cos²θ.
= 1 – 2 sin²θ. cos²θ.

= 1 -	4sin²θ . cos²θ
	2

[∵ sin²θ = 2 sinθ . cosθ]

= 1 -	sin²2θ
	2

= 1 -	1 - cos4θ
	4

[∵ 1 – cos²θ = 2cos²θ]

= 1 -		1	+	cos4θ
		4		4

= 1 -		1	+	1	= 1
		4		4

(cos 4θ ≤1)
OR
The value of sin⁴ θ + cos⁴ θ will be
maximum if θ = 0°
∴ Required value = (sin0)⁴ + (cos0)⁴ = 0 + 1 = 1