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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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