A value of x satisfying the equation sin⁡[ cot^-1⁡(1+x)]= cos⁡[ tan^-1⁡x], is:

If $\sin\left(\dfrac{\pi}{18}\right) \sin\left(\dfrac{5\pi}{18}\right) \sin\left(\dfrac{7\pi}{18}\right) = K$, then the value of $\sin\left(\dfrac{10K\pi}{3}\right)$ is :

If $\frac{\cos ^{2} 48^{\circ}-\sin ^{2} 12^{\circ}}{\sin ^{2} 24^{\circ}-\sin ^{2} 6^{\circ}}=\frac{\alpha+\beta \sqrt{5}}{2}$, where $\alpha, \beta \in \mathbb{N}$, then $\alpha+\beta$ is equal to $\_\_\_\_\_$

The value of sin²30° + cos²30° is:

Considering the principal values of inverse trigonometric functions, the value of the expression $\tan \left(2 \sin ^{-1}\left(\frac{2}{\sqrt{13}}\right)-2 \cos ^{-1}\left(\frac{3}{\sqrt{10}}\right)\right)$ is equal to :

Let $\alpha$ and $\beta$ respectively be the maximum and the minimum values of the function $f(\theta)=4\left(\sin ^{4}\left(\frac{7 \pi}{2}-\theta\right)+\sin ^{4}(11 \pi+\theta)\right)-2\left(\sin ^{6}\left(\frac{3 \pi}{2}-\theta\right)+\sin ^{6}(9 \pi-\theta)\right), \theta \in \mathbf{R}$. Then $\alpha+2 \beta$ is equal to :