Given y = tan−1(

1+cosx

sinx

)
⇒ y = tan−1 (

2cos2 x 2

2sin x 2 .cos x 2

) (Since 1 + cos x = 2 cos2

x

2

and sin x = 2 sin

x

2

cos

x

2

)
⇒ y = tan−1 (cot‌

x

2

)
⇒ y = tan−1 [tan(

π

2

−

x

2

)]
y =

π

2

−

x

2

[Since tan−1 (tan x) = x]
Differentiating with respect to x,
⇒

dy

dx

= 0 -

1

2

= -

1

2