Differentiate `tan^(-1)((sinx)/(1+cosx)),\ -pi

Explanation

Solution:

We use the chain rule to differentiate the expression but it becomes a lot easier to differentiate this expression when we use trigonometric identities.
As given:
`\tan ^{-1}(\frac{1+\cos x}{\sin x})`

We know the following identities:

`\cos x=\frac{1-\tan ^{2}(x / 2)}{1+\tan ^{2}(x / 2)}` and

`\sin x=\frac{2 \tan (x / 2)}{1+\tan ^{2}(x / 2)}`

Substituting these identities for the expression inside the parentheses in the given expression, we get:

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