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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 $${\theta \in}$$ $${[0,\pi]}$$ $${\displaystyle{(1 + {\sin}^2 \theta)(1 + {cos}^2 \theta) < {\frac{9}{4}}}}$$
$${\Leftrightarrow}$$ $${\displaystyle{2 + {{\sin}^2{\theta}}{{\cos}^2{\theta}}}}$$ $${\displaystyle{< {\frac{9}{4}}}}$$
$${\Leftrightarrow}$$ $${\displaystyle{{{\sin}^2{\theta}}{{\cos}^2{\theta}}}}$$ $${\displaystyle{< {\frac{1}{4}}}}$$