Evaluate the following : $(1 + \tan A + \sec A) (1 + \cot A - cosec A) .$

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Solution

Evaluate $(1 + \tan A + \sec A) (1 + \cot A - cosec A) .$
Convert the given expression into $sine and cosine$
$\begin{array}{rcl} (1 + \tan A + \sec A) (1 + \cot A - cosec A) & = & (1 + \frac{\sin A}{\cos A} + \frac{1}{\cos A}) (1 + \frac{\cos A}{\sin A} - \frac{1}{\sin A}) [∵ \tan A = \frac{\sin A}{\cos A}, \sec A = \frac{1}{\cos A}, \cot A = \frac{\cos A}{\sin A}, cosec A = \frac{1}{\sin A}] \\ = & (\frac{\cos A + \sin A + 1}{\cos A}) (\frac{\sin A + \cos A - 1}{\sin A}) \\ = & \frac{{(\sin A + \cos A)}^{2} - 1}{\sin A \cdot \cos A} [∵ (a + b) (a - b) = a^{2} - b^{2}] \\ = & \frac{\sin^{2} A + \cos^{2} A + 2 \sin A \cdot \cos A - 1}{\sin A \cdot \cos A} [∵ {(a + b)}^{2} = a^{2} + b^{2} + 2 a b] \\ = & \frac{1 + 2 \sin A \cdot \cos A - 1}{\sin A \cdot \cos A} \\ = & \frac{2 \sin A \cdot \cos A}{\sin A \cdot \cos A} \\ = & 2 \end{array}$
Hence, the required answer is $(1 + \tan A + \sec A) (1 + \cot A - cosec A) = 2$ .

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