What is the value of `(tan^4 60^@-sin^4 90^@ ) - 2(tan^2 45^@ +3 cos ^2 0^@)`

Answer

To solve the expression \((\tan^4 60^\circ - \sin^4 90^\circ) - 2(\tan^2 45^\circ + 3 \cos^2 0^\circ)\), we will follow these steps:

### Step 1: Calculate \(\tan 60^\circ\)
We know that:
\[ \tan 60^\circ = \sqrt{3} \]
Now, we need to calculate \(\tan^4 60^\circ\):
\[ \tan^4 60^\circ = (\sqrt{3})^4 = 3^2 = 9 \]

### Step 2: Calculate \(\sin 90^\circ\)
We know that:
\[ \sin 90^\circ = 1 \]
Now, we need to calculate \(\sin^4 90^\circ\):
\[ \sin^4 90^\circ = 1^4 = 1 \]

### Step 3: Substitute values into the expression
Now substitute the values we found into the expression:
\[ \tan^4 60^\circ - \sin^4 90^\circ = 9 - 1 = 8 \]

### Step 4: Calculate \(\tan 45^\circ\)
We know that:
\[ \tan 45^\circ = 1 \]
Now, we need to calculate \(\tan^2 45^\circ\):
\[ \tan^2 45^\circ = 1^2 = 1 \]

### Step 5: Calculate \(\cos 0^\circ\)
We know that:
\[ \cos 0^\circ = 1 \]
Now, we need to calculate \(\cos^2 0^\circ\):
\[ \cos^2 0^\circ = 1^2 = 1 \]

### Step 6: Substitute values into the second part of the expression
Now substitute the values into the second part of the expression:
\[ 2(\tan^2 45^\circ + 3 \cos^2 0^\circ) = 2(1 + 3 \cdot 1) = 2(1 + 3) = 2 \cdot 4 = 8 \]

### Step 7: Final calculation
Now we can combine both parts of the expression:
\[ (8) - (8) = 0 \]

### Final Answer
The value of the expression is:
\[ \boxed{0} \]