$\sec A - \cosec A$
Problem: Simplify the expression \(\frac{\sin A \cdot \tan A}{1 - \cos A} - \frac{\cos A \cdot \cot A}{1 - \sin A}\).
Simplifying Trigonometric Terms
First, let's simplify the individual terms of the expression using the definitions \(\tan A = \frac{\sin A}{\cos A}\) and \(\cot A = \frac{\cos A}{\sin A}\).
Term 1 Simplification
Consider the first term: \(\frac{\sin A \cdot \tan A}{1 - \cos A}\)
- Substitute \(\tan A\): \(\frac{\sin A \cdot (\frac{\sin A}{\cos A})}{1 - \cos A} = \frac{\sin^2 A}{\cos A (1 - \cos A)}\)
- Use the identity \(\sin^2 A = 1 - \cos^2 A\): \(\frac{1 - \cos^2 A}{\cos A (1 - \cos A)}\)
- Factor the numerator: \(\frac{(1 - \cos A)(1 + \cos A)}{\cos A (1 - \cos A)}\)
- Cancel the \((1 - \cos A)\) term (assuming \(A \neq n\pi\)): \(\frac{1 + \cos A}{\cos A}\)
- Separate the terms: \(\frac{1}{\cos A} + \frac{\cos A}{\cos A} = \sec A + 1\)
Term 2 Simplification
Consider the second term: \(\frac{\cos A \cdot \cot A}{1 - \sin A}\)
- Substitute \(\cot A\): \(\frac{\cos A \cdot (\frac{\cos A}{\sin A})}{1 - \sin A} = \frac{\cos^2 A}{\sin A (1 - \sin A)}\)
- Use the identity \(\cos^2 A = 1 - \sin^2 A\): \(\frac{1 - \sin^2 A}{\sin A (1 - \sin A)}\)
- Factor the numerator: \(\frac{(1 - \sin A)(1 + \sin A)}{\sin A (1 - \sin A)}\)
- Cancel the \((1 - \sin A)\) term (assuming \(A \neq \frac{\pi}{2} + n\pi\)): \(\frac{1 + \sin A}{\sin A}\)
- Separate the terms: \(\frac{1}{\sin A} + \frac{\sin A}{\sin A} = \cosec A + 1\)
Final Simplification
Now, subtract the simplified second term from the simplified first term:
- \((\sec A + 1) - (\cosec A + 1)\)
- \(\sec A + 1 - \cosec A - 1\)
- \(\sec A - \cosec A\)
The simplified expression is \(\sec A - \cosec A\). This corresponds to Option A.
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