The expression,$\frac{{\tan \,\left( {{\textstyle{{3\,\pi } \over 2}}\,\, - \,\,\alpha } \right)\,\,\,\cos \,\left( {{\textstyle{{3\,\pi } \over 2}}\,\, - \,\,\alpha } \right)}}{{\cos \,(2\,\pi \,\, - \,\alpha )}}$ $+ cos \left( {\alpha \,\, - \,\,\frac{\pi }{2}} \right) \,sin (\pi -\alpha ) + cos (\pi +\alpha ) sin \,\left( {\alpha \,\, - \,\,\frac{\pi }{2}} \right)$ when simplified reduces to :

$1 + \cos 2x + \cos 4x + \cos 6x = $

Suppose $\theta $ and $\phi  (\ne 0)$ are such that $sec\,(\theta  + \phi ),$ $sec\,\theta $ and $sec\,(\theta  - \phi )$ are in $A.P.$ If $cos\,\theta  = k\,cos\,( \frac {\phi }{2})$ for some $k,$ then $k$ is equal to

The value of $sin\,10^o$ $sin\,30^o$ $sin\,50^o$ $sin\,70^o$ is

If $A$ lies in the third quadrant and $3\ tanA - 4 = 0$ , then find the value of $5\ sin\ 2A + 3\  sinA + 4\  cosA$

If $\tan x + \tan \left( {\frac{\pi }{3} + x} \right) + \tan \left( {\frac{{2\pi }}{3} + x} \right) = 3,$ then