Evaluate : `tan7^(@)tan23^(@)tan60^(@)tan67^(@)tan83^(@)+(cot54^(@))/(tan36^(@))+sin20^(@)sec70^(@)-2`

Answer

To evaluate the expression \( \tan 7^\circ \tan 23^\circ \tan 60^\circ \tan 67^\circ \tan 83^\circ + \frac{\cot 54^\circ}{\tan 36^\circ} + \sin 20^\circ \sec 70^\circ - 2 \), we will follow these steps:

### Step 1: Rewrite the tangent terms using complementary angles
We know that \( \tan(90^\circ - \theta) = \cot \theta \). Therefore, we can rewrite some of the tangent terms:
- \( \tan 67^\circ = \tan(90^\circ - 23^\circ) = \cot 23^\circ \)
- \( \tan 83^\circ = \tan(90^\circ - 7^\circ) = \cot 7^\circ \)

Now, we can rewrite the expression:
\[ \tan 7^\circ \tan 23^\circ \tan 60^\circ \cot 23^\circ \cot 7^\circ + \frac{\cot 54^\circ}{\tan 36^\circ} + \sin 20^\circ \sec 70^\circ - 2 \]

### Step 2: Simplify the tangent and cotangent products
From the expression, we can see that:
- \( \tan 23^\circ \) and \( \cot 23^\circ \) will cancel each other out.
- \( \tan 7^\circ \) and \( \cot 7^\circ \) will also cancel each other out.

Thus, we are left with:
\[ \tan 60^\circ + \frac{\cot 54^\circ}{\tan 36^\circ} + \sin 20^\circ \sec 70^\circ - 2 \]

### Step 3: Evaluate \( \tan 60^\circ \)
We know that:
\[ \tan 60^\circ = \sqrt{3} \]

### Step 4: Simplify \( \frac{\cot 54^\circ}{\tan 36^\circ} \)
Using the identity \( \cot \theta = \frac{1}{\tan \theta} \), we have:
\[ \cot 54^\circ = \frac{1}{\tan 54^\circ} \]
And since \( \tan 54^\circ = \cot 36^\circ \), we can write:
\[ \frac{\cot 54^\circ}{\tan 36^\circ} = \frac{1/\tan 54^\circ}{\tan 36^\circ} = \frac{1}{\tan 54^\circ \tan 36^\circ} \]

Using the identity \( \tan(90^\circ - \theta) = \cot \theta \), we find:
\[ \tan 54^\circ = \cot 36^\circ \]
Thus:
\[ \frac{\cot 54^\circ}{\tan 36^\circ} = 1 \]

### Step 5: Simplify \( \sin 20^\circ \sec 70^\circ \)
We know that:
\[ \sec 70^\circ = \frac{1}{\cos 70^\circ} = \frac{1}{\sin 20^\circ} \]
Thus:
\[ \sin 20^\circ \sec 70^\circ = \sin 20^\circ \cdot \frac{1}{\sin 20^\circ} = 1 \]

### Step 6: Combine all parts
Now, substituting back into the expression gives:
\[ \sqrt{3} + 1 + 1 - 2 = \sqrt{3} \]

### Final Answer
Thus, the final answer is:
\[ \sqrt{3} \]