What is the value of the following expression? \(\tan A\left[\frac{\sec A + \tan A}{\operatorname{cosec}A - \cot A}\right]\)
\(\frac{1+\cos A}{1-\sin A}\)
Write the bracket in terms of sin A and cos A: sec A + tan A = \(\frac{1+\sin A}{\cos A}\) and cosec A − cot A = \(\frac{1-\cos A}{\sin A}\).
So the bracket = \(\frac{1+\sin A}{\cos A}\times\frac{\sin A}{1-\cos A}\).
Multiplying by tan A = \(\frac{\sin A}{\cos A}\) gives \(\frac{\sin^{2}A\,(1+\sin A)}{\cos^{2}A\,(1-\cos A)}\).
Now sin2A = 1 − cos2A = (1 − cos A)(1 + cos A) and cos2A = 1 − sin2A = (1 − sin A)(1 + sin A).
Cancelling (1 − cos A) and (1 + sin A) leaves \(\frac{1+\cos A}{1-\sin A}\).
Hence, the answer is \(\frac{1+\cos A}{1-\sin A}\).
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