$2 \sin^2 \beta + 4 \cos(\alpha + \beta) \sin \alpha \sin \beta + \cos 2(\alpha + \beta) = $

જો $\cos \theta - \sin \theta = \sqrt{5} \sin \theta$ હોય,તો $\cos \theta + \sin \theta = $

જો $\cos \frac{\pi}{15} \cos \frac{2 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{5 \pi}{15} \cos \frac{7 \pi}{15} \cos \frac{30 \pi}{15} = x$ હોય,તો $\frac{1}{8x} =$

$\sum\limits_{r = 1}^{89} {{\log _3}(\tan {r^o})} = $

જો $\cos A+\cos (A+B)+\cos (A+2 B)+\ldots$ $n$ પદો સુધી $=$ $\cos \left(\frac{2 A+(n-1) B}{2}\right) \sin \frac{n B}{2} \operatorname{cosec} \frac{B}{2}$ હોય,તો $\cos \frac{\pi}{19}+\cos \frac{3 \pi}{19}+\cos \frac{5 \pi}{19}+\ldots+\cos \frac{17 \pi}{19} = $

$\operatorname{coth}^2 x - \tanh^2 x =$