If $\sin 600^{\circ} \cos 30^{\circ} + \cos 120^{\circ} \sin 150^{\circ} = k$,then $k =$

If $\cos x + \cos^2 x = 1$,then the value of $\sin^2 x + \sin^4 x$ is

If $\frac{{\sin^4 A}}{a} + \frac{{\cos^4 A}}{b} = \frac{1}{{a + b}},$ then the value of $\frac{{\sin^8 A}}{{a^3}} + \frac{{\cos^8 A}}{{b^3}}$ is equal to

$\left( \frac{\sin 2A}{1 + \cos 2A} \right) \left( \frac{\cos A}{1 + \cos A} \right) = $

$\frac{2\sin \theta \tan \theta (1 - \tan \theta ) + 2\sin \theta \sec^2 \theta}{(1 + \tan \theta )^2} = $

If $\tan \theta = \sqrt{\frac{3}{2}}$,the sum of the infinite series $1 + 2(1 - \cos \theta) + 3(1 - \cos \theta)^2 + 4(1 - \cos \theta)^3 + \dots \infty$ is