‌λ=cos22x−2sin‌4x−2cos2x ‌‌ convert all in to ‌cos‌x‌. ‌ ‌λ=(2cos2x−1)2−2(1−cos2x)2−2cos2x ‌=4cos4x−4cos2x+1−2(1−2cos2x+cos4x)− ‌2cos2x ‌=2cos4x−2cos2x+1−2 ‌=2cos4x−2cos2x−1 ‌=2[cos4x−cos2x−‌
1
2
] ‌=2[(cos2x−‌
1
2
)2−‌
3
4
] ‌λmax=2[‌
1
4
−‌
3
4
]=2×(−‌
2
4
)=−1‌ (max Value) ‌ ‌λmin=2[0−‌
3
4
]=−‌
3
2
‌ (MinimumValue) ‌ ‌‌ So, Range ‌=[−‌