(cot θ + tan θ)(cosec θ − sin θ)(cos θ − sec θ) = ________.

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The correct option is -1.

We are given the expression:

$(\cot \theta +\tan \theta )(\theta -\sin \theta )(\cos \theta -\sec \theta )$

Let us express all the trigonometric functions in terms of $\sin \theta$ and $\cos \theta$:

1. Simplify $\cot \theta +\tan \theta$:

$\cot \theta +\tan \theta =\frac{\cos \theta }{\sin \theta }+\frac{\sin \theta }{\cos \theta }=\frac{{\cos }^2\theta +{\sin }^2\theta }{\sin \theta \cos \theta }=\frac{1}{\sin \theta \cos \theta }$

2. Simplify $\theta -\sin \theta$:

$\theta -\sin \theta =\frac{1}{\sin \theta }-\sin \theta =\frac{1-{\sin }^2\theta }{\sin \theta }=\frac{{\cos }^2\theta }{\sin \theta }$

3. Simplify $\cos \theta -\sec \theta$:

$\cos \theta -\sec \theta =\cos \theta -\frac{1}{\cos \theta }=\frac{{\cos }^2\theta -1}{\cos \theta }=\frac{-{\sin }^2\theta }{\cos \theta }$

Now, substitute all simplified terms back into the original expression:

$\left(\frac{1}{\sin \theta \cos \theta }\right)\times \left(\frac{{\cos }^2\theta }{\sin \theta }\right)\times \left(\frac{-{\sin }^2\theta }{\cos \theta }\right)$

$=\frac{-{\sin }^2\theta {\cos }^2\theta }{{\sin }^2\theta {\cos }^2\theta }=-1$

Hence, the value of the given expression is -1.