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Question

The minimum value of (sin2θ+cos2θ+sec2θ+cosec2 θ+tan2θ+cot2θ) is

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Solution

(sin2θ+cos2θ+sec2θ+cosec2 θ+tan2θ+cot2θ)=(1+sec2θ+cosec2 θ+tan2θ+cot2θ)=(1+1+tan2θ+1+cot2θ+tan2θ+cot2θ)=(3+2tan2θ+2cot2θ)=3+2(tan2θ+cot2θ)
The value of above expression is minimum,when tan2θ+cot2θ is minimum.
We know that tan2θ+cot2θ2
So, minimum value is 7.

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