UPSC NDA Math Model Paper 6

Let us put y=√

1+cos‌x

1−cos‌x

.
Using the double angle formula for cosine function we can write:

1+cos‌x=2cos2

x

2

Similarly using the another formula for cosine function we can write:
1−cos‌x=2sin2

x

2

Substituting in value of y we can write:
y=√

2cos2 x 2

2sin2 x 2

=√

cos2 x 2

sin2 x 2

=√cot2

x

2

=cot‌

x

2

Therefore, y=cot‌

x

2

.
Differentiating both sides with respect to x we get,

dy

dx

=−

1

2

cosec2

x

2

Therefore, the final answer is

dy

dx

=−

cosec‌2‌ x 2

2

.