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The integer part of the number $\sum_{\mathrm{k}=0}^{44} \frac{1}{\cos \mathrm{k}^{\circ} \cos (\mathrm{k}+1)^{\circ}} \mathrm{is}$
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The correct answer is:
57
$$
\begin{array}{l}
\frac{1}{\cos 0^{\circ} \cos 1^{\circ}}+\frac{1}{\cos 1^{\circ} \cos 2^{\circ}}+\frac{1}{\cos 2^{\circ} \cos 3^{\circ}}+\ldots \\
\ldots \frac{1}{\cos 44^{\circ} \cos 45^{\circ}}
\end{array}
$$
multiply \& divided by $\sin 1^{\circ}$
$$
\begin{array}{l}
\frac{1}{\sin 1^{\circ}}\left[\frac{\sin 1^{\circ}}{\cos 0^{\circ} \cos 1^{\circ}}+\frac{\sin 1^{\circ}}{\cos 1^{\circ} \cos 2^{\circ}}+\ldots \frac{\sin 1^{\circ}}{\cos 44^{\circ} \cos 45^{\circ}}\right] \\
\frac{1}{\sin 1^{\circ}}\left[\frac{\sin \left(1^{\circ}-0^{\circ}\right)}{\cos 0^{\circ} \cos 1^{\circ}}+\frac{\sin (2-1)^{\circ}}{\cos 1^{\circ} \cos 2^{\circ}}+\ldots \frac{\sin \left(45^{\circ}-44^{\circ}\right)}{\cos 44^{\circ} \cos 45^{\circ}}\right] \\
\frac{1}{\sin 1^{\circ}}\left[\tan 1^{\circ}-\tan 0^{\circ}+\tan 2^{\circ}-\tan 1^{\circ}+\ldots \tan 45^{\circ}-\tan 44^{\circ}\right] \\
=\frac{1}{\sin 1^{\circ}}\left[\tan 45^{\circ}\right] \\
=\frac{1}{0.0174524}=57.2987 \\
\text { Integral part }=57
\end{array}
$$
\begin{array}{l}
\frac{1}{\cos 0^{\circ} \cos 1^{\circ}}+\frac{1}{\cos 1^{\circ} \cos 2^{\circ}}+\frac{1}{\cos 2^{\circ} \cos 3^{\circ}}+\ldots \\
\ldots \frac{1}{\cos 44^{\circ} \cos 45^{\circ}}
\end{array}
$$
multiply \& divided by $\sin 1^{\circ}$
$$
\begin{array}{l}
\frac{1}{\sin 1^{\circ}}\left[\frac{\sin 1^{\circ}}{\cos 0^{\circ} \cos 1^{\circ}}+\frac{\sin 1^{\circ}}{\cos 1^{\circ} \cos 2^{\circ}}+\ldots \frac{\sin 1^{\circ}}{\cos 44^{\circ} \cos 45^{\circ}}\right] \\
\frac{1}{\sin 1^{\circ}}\left[\frac{\sin \left(1^{\circ}-0^{\circ}\right)}{\cos 0^{\circ} \cos 1^{\circ}}+\frac{\sin (2-1)^{\circ}}{\cos 1^{\circ} \cos 2^{\circ}}+\ldots \frac{\sin \left(45^{\circ}-44^{\circ}\right)}{\cos 44^{\circ} \cos 45^{\circ}}\right] \\
\frac{1}{\sin 1^{\circ}}\left[\tan 1^{\circ}-\tan 0^{\circ}+\tan 2^{\circ}-\tan 1^{\circ}+\ldots \tan 45^{\circ}-\tan 44^{\circ}\right] \\
=\frac{1}{\sin 1^{\circ}}\left[\tan 45^{\circ}\right] \\
=\frac{1}{0.0174524}=57.2987 \\
\text { Integral part }=57
\end{array}
$$
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