`(cosec-sinx)(secx-cosx)(tanx+cotx)=?`

Options

A. 0
B. `-1`
C. 1
D. 2

Answer

To solve the expression \((\csc x - \sin x)(\sec x - \cos x)(\tan x + \cot x)\), we will simplify each part step by step.

### Step 1: Rewrite the trigonometric functions
We start by rewriting the cosecant, secant, and cotangent functions in terms of sine and cosine:
\[ \csc x = \frac{1}{\sin x}, \quad \sec x = \frac{1}{\cos x}, \quad \cot x = \frac{\cos x}{\sin x} \]
So, we can express the original expression as:
\[ \left(\frac{1}{\sin x} - \sin x\right)\left(\frac{1}{\cos x} - \cos x\right)\left(\tan x + \frac{\cos x}{\sin x}\right) \]

### Step 2: Simplify each term
Now, we simplify each term:
1. For \(\csc x - \sin x\):
\[ \frac{1 - \sin^2 x}{\sin x} = \frac{\cos^2 x}{\sin x} \]
2. For \(\sec x - \cos x\):
\[ \frac{1 - \cos^2 x}{\cos x} = \frac{\sin^2 x}{\cos x} \]
3. For \(\tan x + \cot x\):
\[ \tan x + \cot x = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} = \frac{1}{\sin x \cos x} \]

### Step 3: Combine the simplified terms
Now we can combine these simplified terms:
\[ \left(\frac{\cos^2 x}{\sin x}\right)\left(\frac{\sin^2 x}{\cos x}\right)\left(\frac{1}{\sin x \cos x}\right) \]

### Step 4: Multiply the fractions
Now we multiply the fractions:
\[ = \frac{\cos^2 x \cdot \sin^2 x \cdot 1}{\sin x \cdot \cos x \cdot \sin x \cdot \cos x} = \frac{\cos^2 x \sin^2 x}{\sin^2 x \cos^2 x} = 1 \]

### Final Answer
Thus, the value of \((\csc x - \sin x)(\sec x - \cos x)(\tan x + \cot x)\) is:
\[ \boxed{1} \]