What is the value of $$\left[(\sec 2\theta + 1)\sqrt{\sec^2 \theta -1}\right] \times \frac{1}{2} (\cot \theta - \tan \theta)$$

we know $$\sqrt{\ \sec^2\theta\ -1}=\tan\theta\ $$
Now $$\frac{1}{2}\left(\cot\ \theta\ -\tan\theta\ \right)=\frac{\frac{1}{2}\left(\cos^2\theta\ -\sin^2\theta\ \right)}{\sin\theta\ \cos\theta\ }=\frac{\cos2\theta}{\sin2\theta\ }$$
So we get 
$$\left(\sec2\theta\ +1\right)\times\ \tan\theta\ \times\ \frac{\cos2\theta\ }{\sin2\theta\ }$$
Multiplying we get :
$$\frac{\tan\theta\ }{\sin2\theta\ }+\frac{\tan\theta\ \cos2\theta\ }{\sin2\theta\ }$$
we get $$\frac{\tan\theta\ }{\sin2\theta\ }\left(1+\cos2\theta\ \right)\ $$
we get $$\frac{\tan\theta\ }{2\sin\theta\ \cos\theta\ }\times\ 2\cos^2\theta\ \ $$
=1