1.

Cos37°+sin37°/Cos37°-sin37° =cot8°

Answer»

LHS = \frac{cos 37\degree + sin 37 \degree }{cos 37\degree - sin 37 \degree} \\= \frac{cos 37\degree + sin (90 - 53) \degree }{cos 37\degree - sin (90- 53) \degree} \\= \frac{cos 37\degree + cos 53 \degree }{cos 37\degree - cos 53 \degree} \\= \frac{ 2 cos \Big( \frac{37+53}{2}\Big) cos \Big( \frac{37-53}{2}\Big) }{ -2 sin \Big( \frac{37+53}{2}\Big) sin \Big( \frac{37-53}{2}\Big) }\\= \frac{ 2 cos \Big( \frac{90}{2}\Big) cos \Big( \frac{-16}{2}\Big) }{ -2 sin \Big( \frac{90}{2}\Big) sin \Big( \frac{-16}{2}\Big) }\\= \frac{ cos 45 \degree cos (-8) \degree }{ - sin 45 \degree sin (-8) \degree } \\= \frac{\frac{1}{\sqrt{2}} \times cos 8 \degree }{ - \frac{1}{\sqrt{2}} \times ( - sin \: 8 \degree ) }\\= \frac{cos 8 \degree }{ sin 8 \degree } \\= cot 8 \degree \\ = RHS

Hence\: proved

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