The maximum value of the expression`1/(sin^2theta+3sinthetacostheta+5cos^2theta` is

Explanation

`f(theta)=sin^2theta+3sinthetacostheta+5cos^2theta`

`=(sin^2theta+cos^2theta)+3/2(2sinthetacostheta)+4cos^2theta`

`=1+3/2sin^2theta+2(2cos^2theta)`

`=1+3/2sin2theta+2(1+cos2theta)`

`f(theta)=2cos2theta+3/2sin2theta+3`

`-sqrt(a^2+b^2)<=asintheta+bcostheta<=sqrt(a^2+b^2)`

`=sqrt(2^2+(3/2)^2)<=2cos2theta+3/2sin2theta<=sqrt(2^2+(3/2)^2)`

`-5/2<=2cos2theta+3/2sin2theta<=11/2`

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