Find the value of :
`3sin^(2)30^(@)+2tan^(2)60^(@)-5cos^(2)45^(@)`

Answer

To solve the expression \( 3\sin^2(30^\circ) + 2\tan^2(60^\circ) - 5\cos^2(45^\circ) \), we will follow these steps:

### Step 1: Calculate \( \sin(30^\circ) \)
We know from trigonometric ratios that:
\[ \sin(30^\circ) = \frac{1}{2} \]

### Step 2: Calculate \( \sin^2(30^\circ) \)
Now, we can find \( \sin^2(30^\circ) \):
\[ \sin^2(30^\circ) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 3: Calculate \( \tan(60^\circ) \)
Next, we find \( \tan(60^\circ) \):
\[ \tan(60^\circ) = \sqrt{3} \]

### Step 4: Calculate \( \tan^2(60^\circ) \)
Now, we can find \( \tan^2(60^\circ) \):
\[ \tan^2(60^\circ) = (\sqrt{3})^2 = 3 \]

### Step 5: Calculate \( \cos(45^\circ) \)
Now, we find \( \cos(45^\circ) \):
\[ \cos(45^\circ) = \frac{1}{\sqrt{2}} \]

### Step 6: Calculate \( \cos^2(45^\circ) \)
Now, we can find \( \cos^2(45^\circ) \):
\[ \cos^2(45^\circ) = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

### Step 7: Substitute the values into the expression
Now we substitute the values we found into the original expression:
\[ 3\sin^2(30^\circ) + 2\tan^2(60^\circ) - 5\cos^2(45^\circ) = 3\left(\frac{1}{4}\right) + 2(3) - 5\left(\frac{1}{2}\right) \]

### Step 8: Simplify the expression
Calculating each term:
\[ 3\left(\frac{1}{4}\right) = \frac{3}{4} \]
\[ 2(3) = 6 \]
\[ 5\left(\frac{1}{2}\right) = \frac{5}{2} \]

Now we combine these results:
\[ \frac{3}{4} + 6 - \frac{5}{2} \]

### Step 9: Convert to a common denominator
The common denominator for \( 4 \) and \( 2 \) is \( 4 \):
\[ 6 = \frac{24}{4} \]
\[ \frac{5}{2} = \frac{10}{4} \]

Now substituting back:
\[ \frac{3}{4} + \frac{24}{4} - \frac{10}{4} = \frac{3 + 24 - 10}{4} = \frac{17}{4} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{17}{4} \]

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