Trigonometric Ratio and Identites

Let $f(\theta ) = 3\left( {{{\sin }^4}\left( {{{3\pi } \over 2} - \theta } \right) + {{\sin }^4}(3\pi + \theta )} \right) - 2(1 - {\sin ^2}2\theta )$ and $S = \left\{ {\theta \in [0,\pi ]:f'(\theta ) = - {{\sqrt 3 } \over 2}} \right\}$. If $4\beta = \sum\limits_{\theta \in S} \theta$, then $f(\beta )$ is equal to

If $u = \sqrt {{a^2}{{\cos }^2}\theta + {b^2}{{\sin }^2}\theta } + \sqrt {{a^2}{{\sin }^2}\theta + {b^2}{{\cos }^2}\theta }$ then the difference between the maximum and minimum values of ${u^2}$ is given by :

If $A = {\sin ^2}x + {\cos ^4}x,$ then for all real $x$:

If  m and M are the minimum and the maximum values of 4 + ${1 \over 2}$ sin² 2x $-$ 2cos⁴ x, x $\in$ R, then M $-$ m is equal to :

The maximum value of 3cos$\theta$ + 5sin $\left( {\theta - {\pi \over 6}} \right)$ for any real value of $\theta$ is :

The set of all values of $\lambda$ for which the equation ${\cos ^2}2x - 2{\sin ^4}x - 2{\cos ^2}x = \lambda$ has a real solution $x$, is :

If $x = \sum\limits_{n = 0}^\infty {{{\left( { - 1} \right)}^n}{{\tan }^{2n}}\theta }$ and $y = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\theta }$ for 0 < $\theta$ < ${\pi \over 4}$, then :

If ${e^{\left( {{{\cos }^2}x + {{\cos }^4}x + {{\cos }^6}x + ...\infty } \right){{\log }_e}2}}$ satisfies the equation t² - 9t + 8 = 0, then the value of ${{2\sin x} \over {\sin x + \sqrt 3 \cos x}}\left( {0 < x < {\pi \over 2}} \right)$ is :

The value of sin 10º sin30º sin50º sin70º is :-

$16\sin (20^\circ )\sin (40^\circ )\sin (80^\circ )$ is equal to :