What is the value of `(1/(sin45^@)-sin45^@) (1/(cos 45^@)-cos45^@)(tan45^@+1/(tan45^@))` ?

Answer

To solve the expression \((1/\sin 45^\circ - \sin 45^\circ)(1/\cos 45^\circ - \cos 45^\circ)(\tan 45^\circ + 1/\tan 45^\circ)\), we will follow these steps:

### Step 1: Calculate the values of \(\sin 45^\circ\), \(\cos 45^\circ\), and \(\tan 45^\circ\).
We know that:
\[ \sin 45^\circ = \cos 45^\circ = \frac{1}{\sqrt{2}}, \quad \tan 45^\circ = 1 \]

### Step 2: Substitute these values into the expression.
Substituting the values we calculated:
\[ (1/\sin 45^\circ - \sin 45^\circ) = \left(1/\left(\frac{1}{\sqrt{2}}\right) - \frac{1}{\sqrt{2}}\right) \]
\[ = \left(\sqrt{2} - \frac{1}{\sqrt{2}}\right) \]
\[ = \left(\frac{2}{\sqrt{2}} - \frac{1}{\sqrt{2}}\right) = \frac{2 - 1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Step 3: Calculate \(1/\cos 45^\circ - \cos 45^\circ\).
Similarly,
\[ (1/\cos 45^\circ - \cos 45^\circ) = \left(1/\left(\frac{1}{\sqrt{2}}\right) - \frac{1}{\sqrt{2}}\right) \]
\[ = \left(\sqrt{2} - \frac{1}{\sqrt{2}}\right) = \frac{1}{\sqrt{2}} \]

### Step 4: Calculate \(\tan 45^\circ + 1/\tan 45^\circ\).
\[ \tan 45^\circ + 1/\tan 45^\circ = 1 + 1 = 2 \]

### Step 5: Combine all parts together.
Now, we can combine all the parts:
\[ \left(\frac{1}{\sqrt{2}}\right) \left(\frac{1}{\sqrt{2}}\right) (2) \]
\[ = \frac{1}{2} \cdot 2 = 1 \]

### Final Answer:
The value of the expression is \(1\).

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