The value of `(sin300^(@).tan330^(@).sec420^(@))/(tan135^(@).sin210^(@).sec315^(@))` is

Options

A. `-1`
B. 1
C. `sqrt(2)`
D. `sqrt(3)`

Explanation

To solve the expression \((\sin 300^\circ \cdot \tan 330^\circ \cdot \sec 420^\circ) / (\tan 135^\circ \cdot \sin 210^\circ \cdot \sec 315^\circ)\), we will evaluate each trigonometric function step by step.

### Step 1: Calculate \(\sin 300^\circ\)
\[ \sin 300^\circ = \sin(360^\circ - 60^\circ) = -\sin 60^\circ \]
Since \(300^\circ\) is in the fourth quadrant where sine is negative, we have:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \implies \sin 300^\circ = -\frac{\sqrt{3}}{2} \]

**Hint:** Use the identity \(\sin(360^\circ - x) = -\sin x\) for angles in the fourth quadrant.

### Step 2: Calculate \(\tan 330^\circ\)
\[ \tan 330^\circ = \tan(360^\circ - 30^\circ) = -\tan 30^\circ \]
Since \(330^\circ\) is also in the fourth quadrant where tangent is negative:
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \implies \tan 330^\circ = -\frac{1}{\sqrt{3}} \]

**Hint:** Use the identity \(\tan(360^\circ - x) = -\tan x\) for angles in the fourth quadrant.

### Step 3: Calculate \(\sec 420^\circ\)
\[ \sec 420^\circ = \sec(360^\circ + 60^\circ) = \sec 60^\circ \]
Since \(420^\circ\) is in the first quadrant:
\[ \sec 60^\circ = \frac{1}{\cos 60^\circ} = \frac{1}{\frac{1}{2}} = 2 \]

**Hint:** Use the identity \(\sec(x + 360^\circ) = \sec x\).

### Step 4: Calculate \(\tan 135^\circ\)
\[ \tan 135^\circ = \tan(180^\circ - 45^\circ) = -\tan 45^\circ \]
Since \(135^\circ\) is in the second quadrant where tangent is negative:
\[ \tan 45^\circ = 1 \implies \tan 135^\circ = -1 \]

**Hint:** Use the identity \(\tan(180^\circ - x) = -\tan x\).

### Step 5: Calculate \(\sin 210^\circ\)
\[ \sin 210^\circ = \sin(180^\circ + 30^\circ) = -\sin 30^\circ \]
Since \(210^\circ\) is in the third quadrant where sine is negative:
\[ \sin 30^\circ = \frac{1}{2} \implies \sin 210^\circ = -\frac{1}{2} \]

**Hint:** Use the identity \(\sin(180^\circ + x) = -\sin x\).

### Step 6: Calculate \(\sec 315^\circ\)
\[ \sec 315^\circ = \sec(360^\circ - 45^\circ) = \sec 45^\circ \]
Since \(315^\circ\) is in the fourth quadrant:
\[ \sec 45^\circ = \frac{1}{\cos 45^\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]

**Hint:** Use the identity \(\sec(360^\circ - x) = \sec x\).

### Step 7: Substitute the values into the expression
Now substituting all the values we calculated:
\[ \frac{(-\frac{\sqrt{3}}{2}) \cdot (-\frac{1}{\sqrt{3}}) \cdot 2}{(-1) \cdot (-\frac{1}{2}) \cdot \sqrt{2}} \]

### Step 8: Simplify the expression
Calculating the numerator:
\[ (-\frac{\sqrt{3}}{2}) \cdot (-\frac{1}{\sqrt{3}}) \cdot 2 = \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{3}} \cdot 2 = 1 \]

Calculating the denominator:
\[ (-1) \cdot (-\frac{1}{2}) \cdot \sqrt{2} = \frac{1}{2} \cdot \sqrt{2} = \frac{\sqrt{2}}{2} \]

Thus, the overall expression simplifies to:
\[ \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

### Final Answer
The value of the expression is \(\sqrt{2}\).

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