Let $f _ { k } ( x ) = \frac { 1 } { k } \left( \sin ^ { k } x + \cos ^ { k } x \right)$ where $x \in R$ and $k \geq 1$. Then $f _ { 4 } ( x ) - f _ { 6 } ( x )$ equals
(1) $\frac { 1 } { 4 }$
(2) $\frac { 1 } { 12 }$
(3) $\frac { 1 } { 6 }$
(4) $\frac { 1 } { 3 }$

If $15 \sin ^ { 4 } \alpha + 10 \cos ^ { 4 } \alpha = 6$, for some $\alpha \in R$, then the value of $27 \sec ^ { 6 } \alpha + 8 \operatorname { cosec } ^ { 6 } \alpha$ is equal to:
(1) 350
(2) 500
(3) 400
(4) 250

If $\sin x + \sin ^ { 2 } x = 1 , x \in \left( 0 , \frac { \pi } { 2 } \right)$, then $\left( \cos ^ { 12 } x + \tan ^ { 12 } x \right) + 3 \left( \cos ^ { 10 } x + \tan ^ { 10 } x + \cos ^ { 8 } x + \tan ^ { 8 } x \right) + \left( \cos ^ { 6 } x + \tan ^ { 6 } x \right)$ is equal to :
(1) 4
(2) 1
(3) 3
(4) 2