If $0 < x < \pi $ and $\cos x + \sin x = \frac{1}{2}$,then $tan \,x$ is
$\frac{{1 - \sqrt 7 }}{4}$
$\;\frac{{4 - \sqrt 7 }}{3}$
$ - \frac{{4 + \sqrt 7 }}{3}$
$\;\frac{{1 + \sqrt 7 }}{4}$
$\sin \left( {\frac{\pi }{{10}}} \right)\sin \left( {\frac{{3\pi }}{{10}}} \right) = $
If $A + B + C = \pi $ and $\cos A = \cos B\,\cos C,$ then $\tan B\,\,\tan C$ is equal to
If $\tan \,(A - B) = 1,\,\,\,\sec \,(A + B) = \frac{2}{{\sqrt 3 }},$ then the smallest positive value of $B$ is
If $\sin x + {\sin ^2}x = 1,$ then ${\cos ^8}x + 2{\cos ^6}x + {\cos ^4}x = $
If $\frac{\sin ^4 x}{2}+\frac{\cos ^4 x}{3}=\frac{1}{5},$ then
$(A)$ $\tan ^2 x=\frac{2}{3}$ $(B)$ $\frac{\sin ^8 x}{8}+\frac{\cos ^8 x}{27}=\frac{1}{125}$
$(C)$ $\tan ^2 x=\frac{1}{3}$ $(D)$ $\frac{\sin ^8 x}{8}+\frac{\cos ^8 x}{27}=\frac{2}{125}$