Find the value of $cosec\theta (1-cos\theta )(cosec\theta +cot\theta )$.

The correct answer is 1.

To evaluate the trigonometric expression:

$cosec\theta (1-cos\theta )(cosec\theta +cot\theta )$

We can express all trigonometric functions in terms of $sin\theta$ and $cos\theta$ using the basic identities:
$cosec\theta =\frac{1}{sin\theta }$
$cot\theta =\frac{cos\theta }{sin\theta }$

Substituting these into the given expression, we get:

$=\frac{1}{sin\theta }(1-cos\theta )\left(\frac{1}{sin\theta }+\frac{cos\theta }{sin\theta }\right)$

Simplify the term in the second parenthesis by finding a common denominator:

$=\frac{1}{sin\theta }(1-cos\theta )\left(\frac{1+cos\theta }{sin\theta }\right)$

Combine the numerators and denominators:

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

Using the algebraic difference of squares formula, $(a-b)(a+b)=a^2-b^2$:

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

By the fundamental Pythagorean trigonometric identity, ${sin}^2\theta +{cos}^2\theta =1$, we know that $1-{cos}^2\theta ={sin}^2\theta$. Substituting this into the numerator:

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

Thus, the simplified value of the expression is 1.