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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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