If $\tan x = \frac{2b}{a - c}$ $(a \ne c)$,$y = a \cos^2 x + 2b \sin x \cos x + c \sin^2 x$ and $z = a \sin^2 x - 2b \sin x \cos x + c \cos^2 x$,then:

If $\tan \alpha = n \tan \beta$ and $\sin \alpha = m \sin \beta,$ then $\frac{m^{2}-1}{n^{2}-1} =$

If $\sin ^{2} \alpha+\sin ^{2} \beta=2,$ then the value of $\cos \left(\frac{\alpha+\beta}{2}\right)$ is

If $(\sin \alpha + \operatorname{cosec} \alpha)^{2} + (\cos \alpha + \sec \alpha)^{2} = K + \tan^{2} \alpha + \cot^{2} \alpha$,then the value of $K$ is

If $x = r \cos \theta \cos \phi$,$y = r \cos \theta \sin \phi$,and $z = r \sin \theta$,then the value of $x^{2} + y^{2} + z^{2}$ is

The numerical value of $\frac{\cos ^{2} 45^{\circ}}{\sin ^{2} 60^{\circ}}+\frac{\cos ^{2} 60^{\circ}}{\sin ^{2} 45^{\circ}}-\frac{\tan ^{2} 30^{\circ}}{\cot ^{2} 45^{\circ}}-\frac{\sin ^{2} 30^{\circ}}{\cot ^{2} 30^{\circ}}$ is