Trigonometric Inequality (Tomato subjective 79)

Problem: Let \({{\theta}_1}\), \({{\theta}_2}\), … , \({{\theta}_{10}}\) be any values in the closed interval \({[0,\pi]}\). Show that
\({F}\) = \( {(1 + {\sin}^2 \theta_1)(1 + {\cos}^2 \theta_1)(1 + {\sin}^2 \theta_2)(1 + {\cos}^2 \theta_2)………(1 + {\sin}^2 \theta_{10})(1 + {\cos}^2 \theta_{10})} \) \({\displaystyle{\le({\frac{9}{4}})^{10}}}\).
What is the maximum value attainable by \({F}\) and at what values of \({{\theta}_1}\), \({{\theta}_2}\), … , \({{\theta}_{10}}\), is the maximum value attained?

Solution:
\({F}\) = \({\displaystyle{(1 + {\sin}^2 \theta_1)(1 + {\cos}^2 \theta_1)(1 + {\sin}^2 \theta_2)(1 + {\cos}^2 \theta_2)………(1 + {\sin}^2 \theta_{10})(1 + {\cos}^2 \theta_{10})}}\)
Now we will show that for any \({\theta \in}\) \({[0,\pi]}\) \({\displaystyle{(1 + {\sin}^2 \theta)(1 + {cos}^2 \theta) < {\frac{9}{4}}}}\)
\({\Leftrightarrow}\) \({\displaystyle{2 + {{\sin}^2{\theta}}{{\cos}^2{\theta}}}}\) \({\displaystyle{< {\frac{9}{4}}}}\)
\({\Leftrightarrow}\) \({\displaystyle{{{\sin}^2{\theta}}{{\cos}^2{\theta}}}}\) \({\displaystyle{< {\frac{1}{4}}}}\)
Now \({\displaystyle{{{\sin}^2{\theta}} + {{\cos}^2{\theta}} = 1}}\)
\({\Rightarrow}\) \({\displaystyle{({{\sin}^2{\theta}} + {{\cos}^2{\theta}})^2 = 1}}\)
\({\Rightarrow}\) \({\displaystyle{{\sin}^4{\theta} + {\cos}^4{\theta} + 2 {{\sin}^2{\theta}{\cos}^2{\theta}} = 1}}\)
\({\Rightarrow}\) \({\displaystyle{4 {{\sin}^2{\theta}}{{\cos}^2{\theta}}}}\) \({\displaystyle{< 1}}\) [ as \({{a^2 + b^2} > {2ab}}\) ]
\({\Rightarrow}\) \({\displaystyle{{{\sin}^2{\theta}}{{\cos}^2{\theta}}}}\) \({\displaystyle{< {\frac{1}{4}}}}\)
So for any \({\theta \in}\) \({[0,\pi]}\) \({\displaystyle{(1 + {\sin}^2 \theta)(1 + {\cos}^2 \theta) {\le} {\frac{9}{4}}}}\)
So \({F {\le} ({\frac{9}{4}})^{10}}\) (proved)
Maximum value attained by \({F}\) is \({({\frac{9}{4}})^{10}}\) and will be attained for \({{\theta_i} = {\frac{\pi}{2}} \pm {\frac{\pi}{4}}}\) for \({i = 1, 2, …., 10}\)