If sinθ + cosθ = √2, where 0° < θ < 90°, find the value of sin⁴θ + cos⁴θ.
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Correct answer: A
Explanation
Has been given :
sinθ + cosθ = √2
According to the formula:
(sinθ + cosθ)² = sin²θ + cos²θ + 2sinθcosθ
(√2)² = 1 + 2sinθcosθ
2 = 1 + 2sinθcosθ
2sinθcosθ = 1
sinθcosθ = 1/2
sin⁴θ + cos⁴θ
= (sin²θ + cos²θ)² − 2sin²θcos²θ
= 1² − 2(1/2)²
= 1 − 2(1/4)
= 1 − 1/2
= 1/2