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For \( 0x\frac{\pi}{2} \), let \( P_{\mathrm{mn}}(x)=m \log _{\cos x}(\sin x)+n \log _{\cos x}(\cot x) \); whère \( m, n \in\{1,2, \ldots, 9\} \)
P
[For example :
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\[
\begin{array}{l}
P_{29}(x)=2 \log _{\cos x}(\sin x)+9 \log _{\cos x}(\cot x) \text { and } \\
\left.P_{77}(x)=7 \log _{\cos x}(\sin x)+7 \log _{\cos x}(\cot x)\right]
\end{array}
\]
On the basis of above information , answer the following questions :
Which of the following is always correct?
(a) \( P_{\mathrm{mn}}(x) \geq m \forall m \geq n \)
(b) \( P_{\mathrm{mn}}(x) \geq n \forall m \geq n \)
(c) \( 2 P_{\mathrm{mn}}(x) \leq n \forall m \leq n \)
(d) \( 2 P_{\mathrm{mn}}(x) \leq m \forall m \leq n \)
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