`DeltaPQR` is right angled at Q.If `mangleR=45^(@)`,then find the value of `(tanP-(1)/(2))`.

Options

A. `((2-sqrt(3)))/(sqrt(3))`
B. `((2sqrt(3)-sqrt(6)))/(2sqrt(2))`
C. `(1)/(2)`
D. `((sqrt(6)-6))/(3sqrt(3))`

Answer

To solve the problem step by step, we start with the given information about triangle PQR.

### Step 1: Understand the triangle
We know that triangle PQR is a right triangle with the right angle at Q. This means that the angles in triangle PQR are:
- Angle P
- Angle Q = 90 degrees
- Angle R = 45 degrees

### Step 2: Find angle P
Since the sum of angles in a triangle is 180 degrees, we can find angle P as follows:
\[ \text{Angle P} = 180^\circ - \text{Angle Q} - \text{Angle R} \]
Substituting the known values:
\[ \text{Angle P} = 180^\circ - 90^\circ - 45^\circ = 45^\circ \]

### Step 3: Calculate tan P
Now that we know that angle P is 45 degrees, we can find the value of \(\tan P\):
\[ \tan P = \tan 45^\circ = 1 \]

### Step 4: Substitute into the expression
We need to find the value of \((\tan P - \frac{1}{2})\):
\[ \tan P - \frac{1}{2} = 1 - \frac{1}{2} \]

### Step 5: Simplify the expression
Now, we simplify the expression:
\[ 1 - \frac{1}{2} = \frac{2}{2} - \frac{1}{2} = \frac{1}{2} \]

### Final Answer
Thus, the value of \((\tan P - \frac{1}{2})\) is:
\[ \frac{1}{2} \]

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