Let $$S=\left\{\theta \in[0,2 \pi]: 8^{2 \sin ^{2} \theta}+8^{2 \cos ^{2} \theta}=16\right\} .$$ Then $$n(s) + \sum\limits_{\theta \in S}^{} {\left( {\sec \left( {{\pi \over 4} + 2\theta } \right)\cos ec\left( {{\pi \over 4} + 2\theta } \right)} \right)} $$ is equal to:

The number of solutions of $$|\cos x|=\sin x$$, such that $$-4 \pi \leq x \leq 4 \pi$$ is :

Let for some real numbers $$\alpha$$ and $$\beta$$, $$a = \alpha - i\beta $$. If the system of equations $$4ix + (1 + i)y = 0$$ and $$8\left( {\cos {{2\pi } \over 3} + i\sin {{2\pi } \over 3}} \right)x + \overline a y = 0$$ has more than one solution, then $${\alpha \over \beta }$$ is equal to

The number of solutions of the equation

$$\cos \left( {x + {\pi \over 3}} \right)\cos \left( {{\pi \over 3} - x} \right) = {1 \over 4}{\cos ^2}2x$$, $$x \in [ - 3\pi ,3\pi ]$$ is :