The value of (cosec x+cot x+1)(sec x-tan x-1) is:

Correct option is A

Given:

$$(\cosec x + \cot x + 1)(\sec x - \tan x - 1)$$​

Concept Used:

$$\cosec x = \frac{1}{\sin x}\\ \ \\\cot x = \frac{\cos x}{\sin x}\\ \ \\\sec x = \frac{1}{\cos x}\\ \ \\\tan x = \frac{\sin x}{\cos x}$$ 

​$$\cos^2x + \sin^2x = 1$$​​

Solution:

$$(\cosec x + \cot x + 1)(\sec x - \tan x - 1)\\ \ \\= (\frac{1}{\sin x} + \frac{\cos x}{\sin x} + 1)( \frac{1}{\cos x} - \frac{\sin x}{\cos x} – 1)\\ \ \\= (\frac{1 + \cos x + \sin x}{\sin x}) (\frac{1 - \sin x - \cos x}{\cos x} )$$​

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

​

​​=$$\frac{(1-\cos^2x - \sin^2x - \cos x \sin x-\cos x \sin x)}{\cos x \sin x}$$  

=$$\frac{-2\cos x \sin x }{\cos x \sin x}$$​

= -2 

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