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Question
Prove that
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Solution
\[\text{ LHS }= \frac{\cos11^\circ + \sin11^\circ}{\cos11^\circ - \sin11^\circ}\]
\[ = \frac{\frac{\cos11^\circ}{\cos11^\circ} + \frac{\sin11^\circ}{\cos11^\circ}}{\frac{\cos11^\circ}{\cos11^\circ} - \frac{\sin11^\circ}{\cos11^\circ}} \left( \text{ Dividing numerator and denominator by }\cos11^\circ \right)\]
\[ = \frac{1 + \tan11^\circ}{1 - \tan11^\circ}\]
\[ = \frac{1 + \tan11^\circ}{1 - 1 \times \tan11^\circ}\]
\[ = \frac{\tan45^\circ + \tan11^\circ}{1 - \tan45^\circ \tan11^\circ} \left(\text{ As }\tan45^\circ = 1 \right)\]
\[ = \tan\left( 45^\circ + 11^\circ \right) \left[\text{ As }\frac{\tan A + \tan B}{1 - \tan A \tan B} = \tan\left( A + B \right) \right]\]
\[ = \tan56^\circ\]
= RHS
Hence proved .
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