The number of distinct real roots of $$\left| {\matrix{ {\sin x} & {\cos x} & {\cos x} \cr {\cos x} & {\sin x} & {\cos x} \cr {\cos x} & {\cos x} & {\sin x} \cr } } \right|\,$$
$$\, = 0$$ in the interval $$ - {\pi \over 4} \le x \le {\pi \over 4}$$ is

The maximum value of $$\left( {\cos {\alpha _1}} \right).\left( {\cos {\alpha _2}} \right).....\left( {\cos {\alpha _n}} \right),$$ under the restrictions
$$0 \le {\alpha _1},{\alpha _2},....,{\alpha _n} \le {\pi \over 2}$$ vand $$\left( {\cot {\alpha _1}} \right).\left( {\cot {\alpha _2}} \right)....\left( {\cot {\alpha _n}} \right) = 1$$ is

If $$\alpha + \beta = \pi /2$$ and $$\beta + \gamma = \alpha ,$$ then $$\tan \,\alpha \,$$ equals

Let $$f\left( \theta \right) = \sin \theta \left( {\sin \theta + \sin 3\theta } \right)$$. Then $$f\left( \theta \right)$$ is