Trigonometric F...

If $$a+b=3-\cos { 4\theta  } $$ and $$a-b=4\sin { 2\theta  } $$, then (ab) is always less then equal to

If $$a=\cos { \alpha +\cos { \beta -\cos { (\alpha +\beta ),\quad b=4\sin { \frac { \alpha  }{ 2 } \sin { \frac { \beta  }{ 2 } \cos { (\frac { \alpha +\beta  }{ 2 } ) }  }  }  }  }  } $$ then a-b=

If $$\csc\theta=\dfrac{x^{2}-y^{2}}{x^{2}+y^{2}}$$ where $$x$$ and $$y$$ are two unequal non-zero real numbers, then the number of real values of $$\theta$$ is 

$$\displaystyle\sum_{r=1}^{n-1} \cos^2\left ( \dfrac{r\pi}{n} \right )=? $$

Consider the curve $$x=1-{ 3t }^{ 2 },y=t-{ 3t }^{ 3 }$$.
If a tangent at point $$(1-{ 3t }^{ 2 },t-{ 3t }^{ 3 })$$ inclined at an angel $$\theta $$ to the positive x-axis and another tangent at point P(-2,2) cuts the curve angels at Q.
The value of $$tan\theta +sec\theta $$ is equal to

If $$\cos \theta = \dfrac{\cos \alpha - \cos \beta}{1 - \cos \alpha \cdot \cos \beta}$$ then $$\tan \left(\dfrac{\theta}{2}\right)$$ is _____

The value of $$\sum _{ r=1 }^{ 10 }{ \cos ^{ 3 }{ \frac { r\pi  }{ 3 }  }  } =$$?

$$\dfrac { \tan\theta  }{ 1-\cot\theta  } +\dfrac { \cot\theta  }{ 1-\tan\theta  } $$ is equal to 

For $$-\frac { \pi  }{ 2 } <\theta <\frac { \pi  }{ 2 } ,\frac { \sin { \theta  } +\sin { 2\theta  }  }{ 1+\cos { \theta  } +\cos { 2\theta  }  } $$ lies in the interval

$$ \operatorname{Tan}\left[\frac{1}{2} \sin ^{-1}\left(\frac{2 a}{1+a^{2}}\right)+\frac{1}{2} \cos ^{-1}\left(\frac{1-a^{2}}{1+a^{2}}\right)\right]= $$