What is $${(1 + \tan \alpha \tan \beta )^2} + {(\tan \alpha - \tan \beta )^2} - {\sec ^2}\beta {\sec ^2}\beta $$ equal to?

If $$\sin \theta + \cos \theta = \sqrt 2 \cos \theta $$, then what is $$(\cos \theta - \sin \theta )$$ equal to?

If $$\sin \theta = - {1 \over 2}$$ and $$\tan \theta = {1 \over {\sqrt 3 }}$$, then in which quadrant does $$\theta$$ lie?

If $$p = \cos ec\theta - \cot \theta $$ and $$q = {(\cos ec\theta + \cot \theta )^{ - 1}}$$, then which one of the following is correct?