If `tan^2 45^@-cos^2 30^@=x.sin 45^@ cos45^@` then find the value of x.

Options

A. `1/2`
B. `1/6`
C. `1/4`
D. `5/6`

Answer

To solve the equation \( \tan^2 45^\circ - \cos^2 30^\circ = x \cdot \sin 45^\circ \cdot \cos 45^\circ \), we will follow these steps:

### Step 1: Calculate \( \tan^2 45^\circ \)
We know that:
\[ \tan 45^\circ = 1 \]
Thus,
\[ \tan^2 45^\circ = 1^2 = 1 \]

**Hint:** Remember that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and for \( 45^\circ \), both sine and cosine are equal.

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

**Hint:** Recall the values of cosine for standard angles.

### Step 3: Substitute the values into the equation
Now, substituting the values we calculated into the equation:
\[ 1 - \frac{3}{4} = x \cdot \sin 45^\circ \cdot \cos 45^\circ \]

### Step 4: Simplify the left side
Calculating the left side:
\[ 1 - \frac{3}{4} = \frac{4}{4} - \frac{3}{4} = \frac{1}{4} \]

### Step 5: Calculate \( \sin 45^\circ \) and \( \cos 45^\circ \)
We know that:
\[ \sin 45^\circ = \cos 45^\circ = \frac{1}{\sqrt{2}} \]

### Step 6: Substitute \( \sin 45^\circ \) and \( \cos 45^\circ \) into the equation
Now, substituting these values into the equation:
\[ \frac{1}{4} = x \cdot \left(\frac{1}{\sqrt{2}}\right) \cdot \left(\frac{1}{\sqrt{2}}\right) \]
This simplifies to:
\[ \frac{1}{4} = x \cdot \frac{1}{2} \]

### Step 7: Solve for \( x \)
To find \( x \), we can multiply both sides by 2:
\[ \frac{1}{4} \cdot 2 = x \]
Thus,
\[ x = \frac{1}{2} \]

### Final Answer
The value of \( x \) is \( \frac{1}{2} \).

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