UPSC NDA I 2025 Mathematics Paper

Given,
The function is y=(1−cos(x))−1, where x≠2nπ and n is an integer.
The given function can be written as:
y=‌

1

1−cos(x)

Using the identity 1−cos(x)=2sin‌2(‌

x

2

), we rewrite the function as:
y=‌

1

2sin‌2(‌ x 2 )

The integral we need to evaluate is:
‌∫‌

dx

2sin‌2(‌ x 2 )

‌‌

1

2

‌∫csc2(‌

x

2

)dx
The integral of csc2(x) is known to be −cot(x), so we get:
‌‌

1

2

(−cot(‌

x

2

))+C
‌−‌

1

2

‌cot(‌

x

2

)+C
The final result is −cot(‌

x

2

)+C