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Prove that (y=4sinθ/2+cosθ)−θ - Mathematics

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Prove that `y=(4sintheta)/(2+costheta)-theta `

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Solution

 `y=(4sintheta)/(2+costheta)-0`

 `dy/(d theta)=((2+costheta)(4costheta)+4sin^2theta)/(2+costheta)^2-1`

`=(8costheta+4cos^2theta+4sin^2theta)/(2costheta)^2-1`

`=(8costheta+4)/(2+costheta)^2-1`

 `=(4costheta-cos^2theta)/(2+costheta)^2`

`dy/(d theta)=(costheta(4-costheta))/(2+costheta)^2`

for increasing `dy/(d theta)>0, theta epsilon(0,pi/2)`

`0<= costheta<=1`

(2+cosθ)2 always greater than 0

So, `dy/(d theta) `

Concept: Simple Problems on Applications of Derivatives
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