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What is the minimum value of `(sin^2theta+cos^4theta)`? |
Answer» Let ` P =sin^2theta + cos^4theta` `=> P = sin^2theta +(1-sin^2theta)^2` `=> P = sin^2theta+1+sin^4 theta -2sin^2theta` `=> P = sin^4theta - sin^2theta +1` `=>P = 1+ sin^2theta(sin^2theta - 1)` `=>P = 1- sin^2theta(1 - sin^2theta)` `=>P = 1- sin^2thetacos^2theta` `=>P = 1-(sin2theta)^2/4` `=> P = 1- 1/4sin^2 2theta` We know, `P` will be minimum when `sin^2 2theta` is maximum. Maximum value of `sin^2 2 theta` is `1`. `:. P_(min) = 1-1/4(1) = 3/4` `=>P_(min) = 3/4` |
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