If the sides of a right-angled triangle are ${\cos2\alpha + \cos2\beta + 2\cos(\alpha + \beta )}$ and ${\sin2\alpha + \sin2\beta + 2\sin(\alpha + \beta )}$,then the length of the hypotenuse is:

Consider the following two statements.
Statement $p$: The value of $\sin 120^\circ$ can be derived by taking $\theta = 240^\circ$ in the equation $2\sin \frac{\theta}{2} = \sqrt{1 + \sin \theta} - \sqrt{1 - \sin \theta}$.
Statement $q$: The angles $A, B, C$ and $D$ of any quadrilateral $ABCD$ satisfy the equation $\cos \left( \frac{1}{2}(A + C) \right) + \cos \left( \frac{1}{2}(B + D) \right) = 0$.
Then the truth values of $p$ and $q$ are respectively:

The value of $2(\sin^6 \theta + \cos^6 \theta) - 3(\sin^4 \theta + \cos^4 \theta) + 1$ is

The value of $\sum_{r=1}^{18} \cos^2(5r)^\circ$,where $x^\circ$ denotes the $x$ degree,is equal to

$3(\sin x-\cos x)^{4}+6(\sin x+\cos x)^{2}+4(\sin ^{6} x+\cos ^{6} x)$ is equal to

For $0 < x \leq \pi$, $\sinh ^{-1}(\cot x)$ is equal to