Let P(x)=1−cos⁡2x+sin⁡2x1+cos⁡2x+sin⁡2x2+1+cot⁡x+cot2⁡x1+tan⁡x+tan2⁡x the minimum value of P(x) equals

P(x)=1−cos⁡2x+sin⁡2x1+cos⁡2x+sin⁡2x2+1+cot⁡x+cot2⁡x1+tan⁡x+tan2⁡x=2sin2⁡x+2sin⁡xcos⁡x2cos2⁡x+2sin⁡xcos⁡x2+1tan2⁡x⋅tan2⁡x+tan⁡x+11+tan⁡x+tan2⁡x=tan2⁡x+cot2⁡x=(tan⁡x−cot⁡x)2+2

Let P(x)=1−cos⁡2x+sin⁡2x1+cos⁡2x+sin⁡2x2+1+cot⁡x+cot2⁡x1+tan⁡x+tan2⁡x the minimum value of P(x) equals

P(x)=1−cos⁡2x+sin⁡2x1+cos⁡2x+sin⁡2x2+1+cot⁡x+cot2⁡x1+tan⁡x+tan2⁡x=2sin2⁡x+2sin⁡xcos⁡x2cos2⁡x+2sin⁡xcos⁡x2+1tan2⁡x⋅tan2⁡x+tan⁡x+11+tan⁡x+tan2⁡x=tan2⁡x+cot2⁡x=(tan⁡x−cot⁡x)2+2

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Let $P (x) = {(\frac{1 - \cos 2 x + \sin 2 x}{1 + \cos 2 x + \sin 2 x})}^{2} + (\frac{1 + \cot x + \cot^{2} x}{1 + \tan x + \tan^{2} x})$ the minimum value of P(x) equals

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answer is B.

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Detailed Solution

$\begin{matrix} P (x) = {(\frac{1 - \cos 2 x + \sin 2 x}{1 + \cos 2 x + \sin 2 x})}^{2} + (\frac{1 + \cot x + \cot^{2} x}{1 + \tan x + \tan^{2} x}) \\ = {(\frac{2 \sin^{2} x + 2 \sin xcos x}{2 \cos^{2} x + 2 \sin xcos x})}^{2} + (\frac{1}{\tan^{2} x} \cdot \frac{\tan^{2} x + \tan x + 1}{1 + \tan x + \tan^{2} x}) \\ = \tan^{2} x + \cot^{2} x \\ = (\tan x - \cot x)^{2} + 2 \end{matrix}$