\(\dfrac{1}{sin^4\left(x+\dfrac{\pi}{2}\right)\left[tan^2x+tan^2\left(\dfrac{\pi}{2}-x\right)+2\right]}=\dfrac{1}{cos^4x\left(tan^2x+cot^2x+2\right)}\)
\(=\dfrac{1}{cos^4x\left(tanx+cotx\right)^2}=\dfrac{1}{cos^4x\left(\dfrac{sinx}{cosx}+\dfrac{cosx}{sinx}\right)^2}\)
\(=\dfrac{1}{cos^4x\left(\dfrac{sin^2x+cos^2x}{sinx.cosx}\right)^2}=\dfrac{1}{cos^4x.\dfrac{1}{sin^2x.cos^2x}}\)
\(=\dfrac{sin^2x}{cos^2x}=tan^2x\)
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