$\sin 15^\circ + \cos 105^\circ = $

$\tan \left(-\frac{23 \pi}{3}\right)-\cot \left(\theta-\frac{13 \pi}{3}\right) =$

If $\sin x = -\frac{24}{25},$ then the value of $\tan x$ is

The value of $\sin ^2 \frac{\pi}{8} + \sin ^2 \frac{3\pi}{8} + \sin ^2 \frac{5\pi}{8} + \sin ^2 \frac{7\pi}{8}$ is

If $\left[1-\cos \left(\frac{\pi}{2}+\alpha\right)+\sin \left(\frac{3 \pi}{2}+\alpha\right)\right]^2+\left[1-\sin \left(\frac{3 \pi}{2}-\alpha\right)-\cos \left(\frac{3 \pi}{2}+\alpha\right)\right]^2=a+b \sin ^2\left(\frac{\pi}{4}+\alpha\right)$,then $a^2+b^2=$

Suppose $ABC$ is a triangle and $D, E$ are points on the sides $AB$ and $AC$ respectively. If $AD : AB = 3 : 5$ and $AE : AC = 2 : 3$,then the ratio of the areas of the triangles $ABC$ and $ADE$ lies in the interval.