# Trigo .. again !

Algebra Level 5

Four real numbers $$a,b,c,$$ and $$d$$ in the interval $$[0,\pi]$$ satisfy the following equations:

\begin{align} \sin a+7\sin b&=4(\sin c+2\sin d) \\ \cos a+7\cos b&=4(\cos c+2\cos d). \end{align}

If

$\frac{7\cos(b-c)}{\cos(a-d)}=m$ $\text{and}$ $(1+\tan1^\circ)(1+\tan2^\circ)(1+\tan3^\circ)\cdots(1+\tan45^\circ)=2^n,$

which of the following statements is correct concerning $$m$$ and $$n?$$

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