The value of `4/(tan^2 60^@)+1/(cos^2 30^@)-sin^2 90^@` is equal to ……….

Answer

To solve the expression \( \frac{4}{\tan^2 60^\circ} + \frac{1}{\cos^2 30^\circ} - \sin^2 90^\circ \), we will follow these steps:

### Step 1: Calculate \( \tan^2 60^\circ \)
We know that:
\[ \tan 60^\circ = \sqrt{3} \]
Thus,
\[ \tan^2 60^\circ = (\sqrt{3})^2 = 3 \]

### Step 2: Substitute \( \tan^2 60^\circ \) into the expression
Now, we substitute this value into the expression:
\[ \frac{4}{\tan^2 60^\circ} = \frac{4}{3} \]

### Step 3: Calculate \( \cos^2 30^\circ \)
We know that:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]
Thus,
\[ \cos^2 30^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

### Step 4: Substitute \( \cos^2 30^\circ \) into the expression
Now, we substitute this value into the expression:
\[ \frac{1}{\cos^2 30^\circ} = \frac{1}{\frac{3}{4}} = \frac{4}{3} \]

### Step 5: Calculate \( \sin^2 90^\circ \)
We know that:
\[ \sin 90^\circ = 1 \]
Thus,
\[ \sin^2 90^\circ = 1^2 = 1 \]

### Step 6: Substitute \( \sin^2 90^\circ \) into the expression
Now, we substitute this value into the expression:
\[ -\sin^2 90^\circ = -1 \]

### Step 7: Combine all parts of the expression
Now we combine all the parts:
\[ \frac{4}{3} + \frac{4}{3} - 1 \]

### Step 8: Simplify the expression
Combine the fractions:
\[ \frac{4}{3} + \frac{4}{3} = \frac{8}{3} \]
Then subtract 1 (which is \(\frac{3}{3}\)):
\[ \frac{8}{3} - \frac{3}{3} = \frac{5}{3} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{5}{3} \]

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