UPSC NDA I 2025 Mathematics Paper

Given,
The function is y=(1−cos(x))−1, where x≠2nπ and n is an integer.
The cosine function has a range of [−1,1], so cos(x) can take values from ( -1 ) to ( 1 ).
The expression 1−cos(x) will take values from:
1−1=0‌‌‌ to ‌‌‌1−(−1)=2
So, (1−cos(x)) takes values in the range (0,2], but x≠2nπ excludes the value 0 .
Since y=‌

1

1−cos(x)

, the reciprocal function will take values in the range [‌

1

2

,∞)
∴ The range of the function is [0.5,∞).