Differentiation of log(tan(x/2))

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Q. By using transformation

z = log tan (\frac{x}{z})

the differential equation

y_{2} + cot x \times y_{1} + 4 y {csc}^{2} x = 0

reduces to

Q. Solve :

\int \frac{log (tan \frac{x}{2})}{sin x} d x

If $x ϵ R$ and $n ϵ I$ , then the determinant $Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} sin (n π) & sin x - cos x & log (tan x) cos x - sin x & cos [(2 n + 1) \frac{π}{2}] & log (cot x) log (cot x) & log (tan x) & tan (n π) \end{matrix} ∣ ∣ ∣ ∣ ∣$

\int \frac{d x}{cos x + \sqrt{3} sin x}

equals

\int_{0}^{π / 2} log (tan x + cot x) d x

is equal to

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