Mathematics

Problem

For

0 < x < \frac{\pi}{2}

, let

P_{mn}(x) = m \log_{\cos x}(\sin x) + n \log_{\cos x}(\cot x)

; where

m, n \in \{1, 2, \dots, 9\}

[For example :

P_{29}(x) = 2 \log_{\cos x}(\sin x) + 9 \log_{\cos x}(\cot x) \text{ and}

P_{77}(x) = 7 \log_{\cos x}(\sin x) + 7 \log_{\cos x}(\cot x)]

On the basis of above information, answer the following questions : The mean proportional of numbers

P_{49}\left(\frac{\pi}{4}\right)

and

P_{94}\left(\frac{\pi}{4}\right)

is equal to a. 4 b. 6 c. 9 d. 10