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Question
Prove that
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Solution
\[\text{ LHS }= \frac{\cos8^\circ - \sin8^\circ}{\cos8^\circ + \sin8^\circ}\]
\[ = \frac{\frac{\cos8^\circ}{\cos8^\circ} - \frac{\sin8^\circ}{\cos8^\circ}}{\frac{\cos8}{\cos8} + \frac{\sin8}{\cos8}} \left( \text{ Dividing numeraor and denominator by }\cos 8^\circ \right)\]
\[ = \frac{1 - \tan8^\circ}{1 + \tan8^\circ}\]
\[ = \frac{1 - \tan8^\circ}{1 + 1 \times \tan8^\circ}\]
\[ = \frac{\tan45^\circ - \tan8^\circ}{1 + \tan45^\circ \tan8^\circ} \left( \text{ As }\tan 45^\circ = 1 \right)\]
\[ = \tan\left( 45^\circ - 8^\circ \right) \left[\text{ As }\frac{\tan A - \tan B}{1 + \tan A \tan B} = \tan\left( A + B \right) \right]\]
\[ = \tan37^\circ\]
= RHS
Hence proved.
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