Prove that

$3 \sin \frac{\pi}{6} \sec \frac{\pi}{3}-4 \sin \frac{5 \pi}{6} \cot \frac{\pi}{4}=1$

If the arcs of the same length in two circles $S_1$ and $S_2$ subtend angles $75^o $ and $120^o $ respectively at the centre. The ratio $\frac{{{S_1}}}{{{S_2}}}$ is equal to

If $x + \frac{1}{x} = 2\cos \alpha $, then ${x^n} + \frac{1}{{{x^n}}} = $

If $sin\theta_1 + sin\theta_2 + sin\theta_3 = 3,$ then $cos\theta_1 + cos\theta_2 + cos\theta_3=$

Let the function $:(0, \pi) \rightarrow R$ be defined by

$f (\theta)=(\sin \theta+\cos \theta)^2+(\sin \theta-\cos \theta)^4$

Suppose the function $f$ has a local minimum at $\theta$ precisely when $\theta \in\left\{\lambda_1 \pi, \ldots, \lambda_{ T } \pi\right\}$, where $0<\lambda_1<\cdots<\lambda_r<1$. Then the value of $\lambda_1+\cdots+\lambda_r$ is. . . . . 

At what time between $10\,\,O'clock$ and $11\,\,O 'clock$ are the two hands of a clock symmetric with respect to the vertical line (give the answer to the nearest second)?