Question

solve both (1) and (2)

Answer 1

1.

$\frac{(cos \theta - sin \theta)\ +\ (cos \theta + sin \theta)}{(cos \theta - sin \theta)\ -\ (cos \theta + sin \theta)} \\ \\ = \frac{cos \theta - sin \theta + cos \theta + sin \theta}{cos \theta - sin \theta - cos \theta - sin \theta} \\ \\ = \frac{cos \theta + cos \theta - sin \theta  + sin \theta}{cos \theta- cos \theta - sin \theta  - sin \theta} \\ \\ = \frac{2\ cos \theta}{ -2\ sin \theta}\\ \\ = -\frac{cos \theta}{ sin \theta} \\ \\ =-cot \theta$

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2.

$\frac{(1-\sqrt{3}) + (1+\sqrt{3}) }{(1-\sqrt{3}) - (1+\sqrt{3})} \\ \\ =\frac{1-\sqrt{3} + 1+\sqrt{3} }{1-\sqrt{3} - 1-\sqrt{3}} \\ \\ =\frac{1+1-\sqrt{3}+\sqrt{3}  }{1-1-\sqrt{3}-\sqrt{3}} \\ \\ =\frac{2}{-2\sqrt{3} } \\ \\ =-\frac{1}{\sqrt{3} }$