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Question Number 87624 by john santu last updated on 05/Apr/20
find minimum value of cos^2 w +  sec^2 w
$$\mathrm{find}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{cos}\:^{\mathrm{2}} \mathrm{w}\:+ \\ $$$$\mathrm{sec}\:^{\mathrm{2}} \mathrm{w}\: \\ $$
Commented by john santu last updated on 05/Apr/20
AM ≥ GM  ((cos^2  w+sec^2  w)/2) ≥ (√(cos^2 w×sec^2 w))  cos^2 w  + sec^2 w ≥ 2  minimum value = 2
$$\mathrm{AM}\:\geqslant\:\mathrm{GM} \\ $$$$\frac{\mathrm{cos}\:^{\mathrm{2}} \:\mathrm{w}+\mathrm{sec}\:^{\mathrm{2}} \:\mathrm{w}}{\mathrm{2}}\:\geqslant\:\sqrt{\mathrm{cos}\:^{\mathrm{2}} \mathrm{w}×\mathrm{sec}\:^{\mathrm{2}} \mathrm{w}} \\ $$$$\mathrm{cos}\:^{\mathrm{2}} \mathrm{w}\:\:+\:\mathrm{sec}\:^{\mathrm{2}} \mathrm{w}\:\geqslant\:\mathrm{2} \\ $$$$\mathrm{minimum}\:\mathrm{value}\:=\:\mathrm{2} \\ $$

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