Find the value of `(sin30^(@).tan330^(@).sec420^(@))/(tan135^(@).sin135^(@).sin210^(@)sec315^(@))`

Explanation

To find the value of the expression \((\sin 30^\circ \cdot \tan 330^\circ \cdot \sec 420^\circ) / (\tan 135^\circ \cdot \sin 135^\circ \cdot \sin 210^\circ \cdot \sec 315^\circ)\), we will evaluate each trigonometric function step by step.

### Step 1: Calculate \(\sin 30^\circ\)
\[ \sin 30^\circ = \frac{1}{2} \]

**Hint:** Remember that \(\sin 30^\circ\) is a well-known value.

### Step 2: Calculate \(\tan 330^\circ\)
\[ \tan 330^\circ = \tan(360^\circ - 30^\circ) = -\tan 30^\circ = -\frac{1}{\sqrt{3}} \]

**Hint:** Use the identity \(\tan(360^\circ - \theta) = -\tan \theta\).

### Step 3: Calculate \(\sec 420^\circ\)
\[ \sec 420^\circ = \sec(360^\circ + 60^\circ) = \sec 60^\circ = 2 \]

**Hint:** Remember that \(\sec \theta = \frac{1}{\cos \theta}\) and that angles greater than \(360^\circ\) can be reduced by subtracting \(360^\circ\).

### Step 4: Calculate \(\tan 135^\circ\)
\[ \tan 135^\circ = \tan(180^\circ - 45^\circ) = -\tan 45^\circ = -1 \]

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

### Step 5: Calculate \(\sin 135^\circ\)
\[ \sin 135^\circ = \sin(180^\circ - 45^\circ) = \sin 45^\circ = \frac{1}{\sqrt{2}} \]

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

### Step 6: Calculate \(\sin 210^\circ\)
\[ \sin 210^\circ = \sin(180^\circ + 30^\circ) = -\sin 30^\circ = -\frac{1}{2} \]

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

### Step 7: Calculate \(\sec 315^\circ\)
\[ \sec 315^\circ = \sec(360^\circ - 45^\circ) = \sec 45^\circ = \sqrt{2} \]

**Hint:** Remember that \(\sec \theta = \frac{1}{\cos \theta}\) and that angles in the fourth quadrant have positive secant.

### Step 8: Substitute all values into the expression
Now substituting all the calculated values into the expression:
\[ \frac{\left(\frac{1}{2}\right) \cdot \left(-\frac{1}{\sqrt{3}}\right) \cdot 2}{\left(-1\right) \cdot \left(\frac{1}{\sqrt{2}}\right) \cdot \left(-\frac{1}{2}\right) \cdot \sqrt{2}} \]

### Step 9: Simplify the expression
Calculating the numerator:
\[ \text{Numerator} = \frac{1}{2} \cdot -\frac{1}{\sqrt{3}} \cdot 2 = -\frac{1}{\sqrt{3}} \]

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

### Step 10: Final calculation
Now we have:
\[ \frac{-\frac{1}{\sqrt{3}}}{\frac{1}{2}} = -\frac{2}{\sqrt{3}} \]

### Final Answer
Thus, the value of the expression is:
\[ -\frac{2}{\sqrt{3}} \]

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