Trigonometric F...

The number of solutions of $$\displaystyle \sum_{r=1}^{5} \cos r x=5 $$ in the interval of $$\displaystyle \left [ 0,2\pi  \right ]$$ is

If $$3x = \text{cosec } \theta$$ and $$\dfrac {3}{x} = \cot \theta$$, then $$\left (x^{2} - \dfrac {1}{x^{2}}\right ) =$$

If $$\displaystyle \sin x+\mathrm{cosec}\: x=2, $$ then $$\displaystyle \sin ^{n} x + \mathrm{cosec} ^{n} \: x$$ is equal to 

The value of $$\displaystyle \frac { \sin { \theta  } \cos { \theta  } .\sin { \left( { 90 }^{ o }-\theta  \right)  }  }{ \cos { \left( { 90 }^{ o }-\theta  \right)  }  } +\frac { \cos { \theta  } .\sin { \theta  } .\cos { \left( { 90 }^{ o }-\theta  \right)  }  }{ \sin { \left( { 90 }^{ o }-\theta  \right)  }  } +\frac { { \sin }^{ 2 }{ 27 }^{ o }+{ \sin }^{ 2 }{ 63 }^{ o } }{ { \cos }^{ 2 }{ 40 }^{ o }+{ \cos }^{ 2 }{ 50 }^{ o } } $$ is :

If $$\displaystyle p=\sqrt{\frac{1-\sin x}{1+\sin x}},q=\frac{1-\sin x}{\cos x},r=\frac{\cos x}{1+\sin x}$$ 
Which one of the following statement is correct ?

The value of $$\alpha \varepsilon (- \pi, 0)$$ satisfying $$sin \alpha + \int_{\alpha}^{2 \alpha} . cos 2x dx = 0$$ is

$$\dfrac12\sin{(2x)}(1+\cot ^{ 2 }{ (x) } )$$ is equal to

The number of $$x\epsilon [0, 2\pi]$$ for which $$|\sqrt {2\sin^{4} x + 18\cos^{2}x} - \sqrt {2\cos^{4} x + 18\sin^{2} x}| = 1$$ is:

The range of $$f(x)=\cfrac { 1 }{ |\sin\ x| } +\cfrac { 1 }{ |\cos\ x| } $$ is

$$\cos (2001) \pi + \cot (2001)\dfrac {\pi}{2} + \sec (2001) \dfrac {\pi}{3} + \tan (2001) \dfrac {\pi}{4} + cosec (2001) \dfrac {\pi}{6}$$ equal to