Evaluate `(sec^2""54^@-cot^2""36^@)/(cosec^2""57^@-tan^2""33^@)+2sin^2""38sec^2""52^@-sin^2""45^@+2/sqrt3tan17^@tan60^@tan73^@`

Answer

To evaluate the expression

\[ \frac{\sec^2 54^\circ - \cot^2 36^\circ}{\csc^2 57^\circ - \tan^2 33^\circ} + 2\sin^2 38^\circ \sec^2 52^\circ - \sin^2 45^\circ + \frac{2}{\sqrt{3}} \tan 17^\circ \tan 60^\circ \tan 73^\circ, \]

we will simplify each term step by step.

### Step 1: Simplify \(\sec^2 54^\circ - \cot^2 36^\circ\)

Using the identity \(\cot(90^\circ - \theta) = \tan \theta\):
\[ \cot 36^\circ = \tan(90^\circ - 36^\circ) = \tan 54^\circ. \]
Thus,
\[ \cot^2 36^\circ = \tan^2 54^\circ. \]
Now, we can rewrite the first part:
\[ \sec^2 54^\circ - \cot^2 36^\circ = \sec^2 54^\circ - \tan^2 54^\circ. \]
Using the identity \(\sec^2 \theta - \tan^2 \theta = 1\):
\[ \sec^2 54^\circ - \tan^2 54^\circ = 1. \]

### Step 2: Simplify \(\csc^2 57^\circ - \tan^2 33^\circ\)

Using the identity \(\tan(90^\circ - \theta) = \cot \theta\):
\[ \tan 33^\circ = \cot(90^\circ - 33^\circ) = \cot 57^\circ. \]
Thus,
\[ \tan^2 33^\circ = \cot^2 57^\circ. \]
Now, we can rewrite the second part:
\[ \csc^2 57^\circ - \tan^2 33^\circ = \csc^2 57^\circ - \cot^2 57^\circ. \]
Using the identity \(\csc^2 \theta - \cot^2 \theta = 1\):
\[ \csc^2 57^\circ - \cot^2 57^\circ = 1. \]

### Step 3: Substitute back into the expression

Now substituting back into the expression:
\[ \frac{1}{1} + 2\sin^2 38^\circ \sec^2 52^\circ - \sin^2 45^\circ + \frac{2}{\sqrt{3}} \tan 17^\circ \tan 60^\circ \tan 73^\circ. \]

### Step 4: Simplify \(2\sin^2 38^\circ \sec^2 52^\circ\)

Using the identity \(\sec^2 \theta = \frac{1}{\cos^2 \theta}\):
\[ \sec^2 52^\circ = \frac{1}{\cos^2 52^\circ}. \]
Thus,
\[ 2\sin^2 38^\circ \sec^2 52^\circ = 2\sin^2 38^\circ \cdot \frac{1}{\cos^2 52^\circ}. \]

### Step 5: Simplify \(-\sin^2 45^\circ\)

We know that:
\[ \sin^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}. \]

### Step 6: Simplify \(\frac{2}{\sqrt{3}} \tan 17^\circ \tan 60^\circ \tan 73^\circ\)

We know that \(\tan 60^\circ = \sqrt{3}\):
\[ \frac{2}{\sqrt{3}} \tan 17^\circ \cdot \sqrt{3} \cdot \tan 73^\circ = 2 \tan 17^\circ \tan 73^\circ. \]
Using the identity \(\tan(90^\circ - \theta) = \cot \theta\):
\[ \tan 73^\circ = \cot 17^\circ. \]
Thus,
\[ \tan 17^\circ \tan 73^\circ = 1. \]
So,
\[ 2 \tan 17^\circ \tan 73^\circ = 2. \]

### Final Step: Combine all parts

Now we combine all parts:
\[ 1 + 2\sin^2 38^\circ \cdot \frac{1}{\cos^2 52^\circ} - \frac{1}{2} + 2. \]
This simplifies to:
\[ 1 + 2 - \frac{1}{2} + 2\sin^2 38^\circ \cdot \frac{1}{\cos^2 52^\circ} = 3 - \frac{1}{2} + 2\sin^2 38^\circ \cdot \frac{1}{\cos^2 52^\circ}. \]

### Conclusion

The final answer can be calculated based on the values of \(\sin^2 38^\circ\) and \(\cos^2 52^\circ\). However, the main evaluation leads us to:

\[ \frac{9}{2}. \]