The value of `sin^(2)30^(@).cos^(2)45^(@)+2tan^(2)30^(@)-sec^(2)60^(@)` is equal to :

Options

A. a.`-(13)/(12)`
B. b.`-(77)/(24)`
C. c.`-(25)/(12)`
D. d.`-(1)/(12)`

Answer

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

### Step 1: Calculate \( \sin^2 30^\circ \)
We know that:
\[ \sin 30^\circ = \frac{1}{2} \]
Thus,
\[ \sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

**Hint:** Remember that \( \sin 30^\circ \) is a standard value in trigonometry.

### Step 2: Calculate \( \cos^2 45^\circ \)
We know that:
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \]
Thus,
\[ \cos^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

**Hint:** The cosine of 45 degrees is also a standard value, often memorized as \( \frac{1}{\sqrt{2}} \).

### Step 3: Calculate \( \tan^2 30^\circ \)
We know that:
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \]
Thus,
\[ \tan^2 30^\circ = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} \]

**Hint:** The tangent of 30 degrees is another standard value that can be derived from sine and cosine.

### Step 4: Calculate \( \sec^2 60^\circ \)
We know that:
\[ \sec 60^\circ = \frac{1}{\cos 60^\circ} = \frac{1}{\frac{1}{2}} = 2 \]
Thus,
\[ \sec^2 60^\circ = 2^2 = 4 \]

**Hint:** Remember that secant is the reciprocal of cosine.

### Step 5: Substitute the values into the expression
Now we substitute the calculated values into the expression:
\[ \sin^2 30^\circ \cdot \cos^2 45^\circ + 2 \tan^2 30^\circ - \sec^2 60^\circ \]
Substituting gives:
\[ \frac{1}{4} \cdot \frac{1}{2} + 2 \cdot \frac{1}{3} - 4 \]

### Step 6: Simplify the expression
Calculating each term:
1. \( \frac{1}{4} \cdot \frac{1}{2} = \frac{1}{8} \)
2. \( 2 \cdot \frac{1}{3} = \frac{2}{3} \)

Now, we have:
\[ \frac{1}{8} + \frac{2}{3} - 4 \]

### Step 7: Find a common denominator and combine
The common denominator for \( 8 \), \( 3 \), and \( 1 \) (for \( 4 \)) is \( 24 \):
1. Convert \( \frac{1}{8} \) to \( \frac{3}{24} \)
2. Convert \( \frac{2}{3} \) to \( \frac{16}{24} \)
3. Convert \( 4 \) to \( \frac{96}{24} \)

Now, substituting gives:
\[ \frac{3}{24} + \frac{16}{24} - \frac{96}{24} = \frac{3 + 16 - 96}{24} = \frac{19 - 96}{24} = \frac{-77}{24} \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{-\frac{77}{24}} \]