If A,B,C are angles of a triangle, then `2sin(A/2)cosec (B/2)sin(C/2)-sinAcot(B/2)-cosA` is (a)independent of A,B,C (b) function of A,B (c)function of C (d) none of these

Explanation

To solve the problem, we need to simplify the expression \(2\sin\left(\frac{A}{2}\right)\csc\left(\frac{B}{2}\right)\sin\left(\frac{C}{2}\right) - \sin A \cot\left(\frac{B}{2}\right) - \cos A\).

### Step 1: Rewrite the expression
The given expression is:
\[ E = 2\sin\left(\frac{A}{2}\right)\csc\left(\frac{B}{2}\right)\sin\left(\frac{C}{2}\right) - \sin A \cot\left(\frac{B}{2}\right) - \cos A \]
We know that \(\csc\left(\frac{B}{2}\right) = \frac{1}{\sin\left(\frac{B}{2}\right)}\) and \(\cot\left(\frac{B}{2}\right) = \frac{\cos\left(\frac{B}{2}\right)}{\sin\left(\frac{B}{2}\right)}\).
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