If $\tan \,(A - B) = 1,\,\,\,\sec \,(A + B) = \frac{2}{{\sqrt 3 }},$ then the smallest positive value of $B$ is

Prove that:   

$\sin ^{2} \frac{\pi}{6}+\cos ^{2} \frac{\pi}{3}-\tan ^{2} \frac{\pi}{4}=-\frac{1}{2}$

Which of the following relations is correct

If $a\,{\cos ^3}\alpha + 3a\,\cos \alpha \,{\sin ^2}\alpha = m$ and $a\,{\sin ^3}\alpha + 3a\,{\cos ^2}\alpha \sin \alpha = n,$ then  ${(m + n)^{2/3}} + {(m - n)^{2/3}}$ is equal to

Prove that: $\sin 3 x+\sin 2 x-\sin x=4 \sin x \cos \frac{x}{2} \cos \frac{3 x}{2}$

If $\sin \theta + \cos \theta = m$ and $\sec \theta + {\rm{cosec}}\theta = n$, then $n(m + 1)(m - 1) = $