`Delta`PQR is right angled at Q. If `m/_ R = 60^@` , then find the value of `(secP - 1//sqrt3)`.

Options

A. `(sqrt6-6)//3sqrt3`
B. `(1-3sqrt2)//3`
C. `1//sqrt3`
D. `2//sqrt3`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the angles in triangle PQR
Given that triangle PQR is right-angled at Q and angle R is 60 degrees, we can find angle P using the angle sum property of triangles.

**Calculation:**
\[ \text{Angle P} + \text{Angle Q} + \text{Angle R} = 180^\circ \]
\[ \text{Angle P} + 90^\circ + 60^\circ = 180^\circ \]
\[ \text{Angle P} + 150^\circ = 180^\circ \]
\[ \text{Angle P} = 180^\circ - 150^\circ = 30^\circ \]

### Step 2: Find sec P
The secant function is defined as the reciprocal of the cosine function:
\[ \sec P = \frac{1}{\cos P} \]
For angle P = 30 degrees:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]
Thus,
\[ \sec P = \frac{1}{\cos 30^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]

### Step 3: Calculate sec P - \(\frac{1}{\sqrt{3}}\)
Now, we need to find:
\[ \sec P - \frac{1}{\sqrt{3}} = \frac{2}{\sqrt{3}} - \frac{1}{\sqrt{3}} \]
Combine the fractions:
\[ \sec P - \frac{1}{\sqrt{3}} = \frac{2 - 1}{\sqrt{3}} = \frac{1}{\sqrt{3}} \]

### Final Answer
The value of \(\sec P - \frac{1}{\sqrt{3}}\) is:
\[ \frac{1}{\sqrt{3}} \]

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