If $x+y=\frac{\pi}{2}$,then the maximum value of $\sin x \cdot \sin y$ is

If $A$ and $B$ are complementary angles,then:

The maximum value of $5 \cos \theta + 3 \cos \left( \theta + \frac{\pi}{3} \right) - 1$ is

The maximum value of the function $f(x) = a \sin x + b \cos x$ is:

If $ABC$ is not a right-angled triangle and $\sin \left(\frac{\pi}{4}-A\right) \sin \left(\frac{\pi}{4}-B\right) = -\frac{1}{2 \sqrt{2}} \operatorname{cosec}\left(\frac{\pi}{4}-C\right)$,then $\tan A \tan B + \tan B \tan C + \tan C \tan A = $

The maximum value of $\cos \alpha_1 \cdot \cos \alpha_2 \cdots \cos \alpha_n$ under the restrictions $0 \le \alpha_1, \alpha_2, \dots, \alpha_n \le \frac{\pi}{2}$ and $\cot \alpha_1 \cdot \cot \alpha_2 \cdots \cot \alpha_n = 1$ is