Differentiate tan⁻¹(1 + cosx/sinx) with respect to x.

One could use the chain rule to differentiate the expression but it becomes a lot easier to differentiate this expression when we use trigonometric identities.
The given expression is:
tan⁻¹(1 + cosx/sinx)
We know the following identities:
cosx = (1 - tan²(x/2))/(1 + tan²(x/2))
sinx = (2tan(x/2))/(1 + tan²(x/2))
Substituting these identities for the expression inside the parentheses in the given expression, we get:
1 + cosx/sinx = 1 + (1 - tan²(x/2))/(1 + tan²(x/2)) / (2tan(x/2))/(1 + tan²(x/2))
= 1 + (1 - tan²(x/2))/(2tan(x/2))
= (2tan(x/2) + 1 - tan²(x/2))/(2tan(x/2))
This expression is still complicated. Let's try another approach.
1 + cosx/sinx = (sinx + cosx)/sinx = (√2(cosx cos(π/4) + sinx sin(π/4)))/sinx = (√2 cos(x - π/4))/sinx
This is still not easy to simplify further.
Let's go back to the original expression:
1 + cosx/sinx = (sinx + cosx)/sinx = cos(x - π/4)/sin x
Let's use a different approach.
1 + cosx/sinx = 1 + cot x = (sinx + cosx)/sinx
We have: 1 + cosx/sinx = (sinx + cosx)/sinx
Let's use the sum-to-product formula:
sinx + cosx = √2 sin(x + π/4)
Therefore,
1 + cosx/sinx = √2 sin(x + π/4)/sinx
This is still difficult to differentiate directly.
Let's try another approach using the identities:
cosx = 1 - 2sin²(x/2)
sinx = 2sin(x/2)cos(x/2)
1 + cosx/sinx = 1 + (1 - 2sin²(x/2))/(2sin(x/2)cos(x/2)) = (2sin(x/2)cos(x/2) + 1 - 2sin²(x/2))/(2sin(x/2)cos(x/2))
This expression is complicated to differentiate.
Let's use a simpler approach. We have:
1 + cosx/sinx = (sinx + cosx)/sinx
Dividing the numerator and denominator by cosx, we get:
(tanx + 1)/tanx = 1 + cotx = cot(x/2)
So, the given expression becomes:
tan⁻¹(cot(x/2)) = tan⁻¹(tan(π/2 - x/2)) = π/2 - x/2
So, differentiating the given expression with respect to x, we get:
d/dx(tan⁻¹(1 + cosx/sinx)) = d/dx(π/2 - x/2) = -1/2