sin²θ + cos²θ + sec²θ + 2cosec²θ + tan²θ

(? sin²θ + cos²θ = 1)

= 1 + sec²θ + 2cosec²θ + tan²θ

= 1 + tan²θ + sec²θ + 2cosec²θ

= 2sec²θ + 2cosec²θ

$(= \;\frac{2}{{{{\sin }^2}\theta }}\; + \;\frac{2}{{{{\cos }^2}\theta }}\; = \;\frac{{2\left( {{{\cos }^2}\theta \; + \;{{\sin }^2}\theta } \right)}}{{{{\cos }^2}\theta .{{\sin }^2}\theta }}\; = \;\frac{2}{{{{\left( {\cos \theta .\sin \theta } \right)}^2}}})$

The given expression will be minimum when the denominator will be maximum.

Maxmium value of (sinθ.cosθ)² = ¼

Minimum value of $(\frac{2}{{{{\left( {\cos \theta .\sin \theta } \right)}^2}}}\; = \;\frac{2}{{\frac{1}{4}}}\; = \;8)$