If $\frac{\cos x}{a} = \frac{\cos (x + \theta)}{b} = \frac{\cos (x + 2\theta)}{c} = \frac{\cos (x + 3\theta)}{d}$,then $\left( \frac{a + c}{b + d} \right)$ is equal to:

If $A+B+C+D=2 \pi$,then $\sin A+\sin B+\sin C+\sin D=$

The value of the expression $\frac{1+\sin 2 \alpha}{\cos (2 \alpha-2 \pi) \tan \left(\alpha-\frac{3 \pi}{4}\right)} - \frac{1}{4} \sin 2 \alpha \left[\cot \frac{\alpha}{2}+\cot \left(\frac{3 \pi}{2}+\frac{\alpha}{2}\right)\right]$ is:

If $A+B+C+D=2 \pi$,then $\cos A-\cos B+\cos C-\cos D=$

If $x \cos \theta + y \sin \theta = 5$ and $x \sin \theta - y \cos \theta = 3$,then the value of $x^{2} + y^{2}$ is:

If $\sin x \cdot \cosh y = \cos \theta$ and $\cos x \cdot \sinh y = \sin \theta$,then $\sin^2 x + \cosh^2 y = $