Let ${f_k}(x) = \frac{1}{k}(\sin^k x + \cos^k x)$ for $k = 1, 2, 3, ...$. Then for all $x \in R$,the value of $f_4(x) - f_6(x)$ is equal to

The value of $\frac{\sin 150^{\circ}-5 \cos 300^{\circ}+7 \tan 225^{\circ}}{\tan 135^{\circ}+3 \sin 210^{\circ}}$ is

If $A + B + C = \pi$ $(A, B, C > 0)$ and the angle $C$ is obtuse,then:

In triangle $ABC$,the value of $\sin 2A + \sin 2B + \sin 2C$ is equal to

The value of $\cos y \cos \left( \frac{\pi}{2} - x \right) - \cos \left( \frac{\pi}{2} - y \right) \cos x + \sin y \cos \left( \frac{\pi}{2} - x \right) + \cos x \sin \left( \frac{\pi}{2} - y \right)$ is zero,if

If $\frac{\sec 8\theta - 1}{\sec 4\theta - 1} = \frac{a + b\tan^2 2\theta}{1 + c\tan^2 2\theta + d\tan^4 2\theta}$ (where $\theta \neq \frac{n\pi}{16}, n \in I$),then the value of $(a - b + c - d)$ is -