Which of the following relations is correct?

If $A = 130^\circ$ and $x = \sin A + \cos A$,then

If $A$ lies in the third quadrant and $3\tan A - 4 = 0,$ then $5\sin 2A + 3\sin A + 4\cos A = $

If $5 \cos \theta + 12 \sin \theta = 13$ and $0^{\circ} < \theta < 90^{\circ}$,then the value of $\sin \theta$ is:

If $\sec \alpha + \tan \alpha = 2,$ then the value of $\sin \alpha$ is (assume that $0 < \alpha < 90^{\circ}$).

If $\tan \alpha = \frac{x^2 - x}{x^2 - x + 1}$ and $\tan \beta = \frac{1}{2x^2 - 2x + 1}$ $(x \ne 0, 1)$,where $0 < \alpha, \beta < \frac{\pi}{2}$,then $\tan(\alpha + \beta)$ has the value equal to: