The value of `("cosec"a-sina)(seca-cosa)(tana+cota)` is

Options

A. 1
B. 2
C. 6
D. 4

Answer

To find the value of the expression \((\csc \alpha - \sin \alpha)(\sec \alpha - \cos \alpha)(\tan \alpha + \cot \alpha)\), we can evaluate it by substituting a specific angle for \(\alpha\). A good choice is to use \(\alpha = 45^\circ\) because the trigonometric values at this angle are straightforward.

### Step-by-Step Solution:

1. **Substitute \(\alpha = 45^\circ\)**:
\[ \csc 45^\circ = \frac{1}{\sin 45^\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]
\[ \sin 45^\circ = \frac{1}{\sqrt{2}} \]
\[ \sec 45^\circ = \frac{1}{\cos 45^\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \]
\[ \tan 45^\circ = 1 \]
\[ \cot 45^\circ = 1 \]

2. **Calculate each part of the expression**:
- First part:
\[ \csc 45^\circ - \sin 45^\circ = \sqrt{2} - \frac{1}{\sqrt{2}} = \sqrt{2} - \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} - \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]
- Second part:
\[ \sec 45^\circ - \cos 45^\circ = \sqrt{2} - \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} - \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]
- Third part:
\[ \tan 45^\circ + \cot 45^\circ = 1 + 1 = 2 \]

3. **Combine the results**:
\[ (\csc 45^\circ - \sin 45^\circ)(\sec 45^\circ - \cos 45^\circ)(\tan 45^\circ + \cot 45^\circ) = \left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{\sqrt{2}}\right)(2) \]
\[ = \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}\right) \cdot 2 = \left(\frac{1}{2}\right) \cdot 2 = 1 \]

Thus, the value of the expression is \(\boxed{1}\).