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# 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 $${i = 1, 2, …., 10}$$

August 3, 2015