UPSC CDS 1 2022 Math Paper

Given:
Trigonometric Expression is
sin4θ+cos4θ−2sin2θcos2θ
Concept Used:
‌‌−1≤cos‌θ≤1,0≤cos2θ≤1
‌‌−1≤sin‌θ≤1,0≤sin2θ≤1
cos2θ−sin2θ=cos‌2‌θ
Calculation:
We have sin4θ+cos4θ−2sin2θcos2θ
⇒(cos2θ)2+(sin2θ)2−2⋅sin2θ⋅cos2θ
Using formula a2+b2−2ab=(a−b)2
⇒(cos2θ−sin2θ)2
⇒(cos‌2‌θ)2
⇒cos22θ
According to the concept used
The minimum value of cos‌θ is 0
∴ The minimum value of cos22θ is 0 .