If `((1)/(costheta))-((1)/(cottheta))=(1)/(P)` , then what is the valueof `costheta` ?

Options

A. `((P+1))/(P-1)`
B. `((P^(2)+1))/(2P)`
C. `(2(P^(2)+1))/(P)`
D. `(2P)/((P^(2)+1))`

Answer

To solve the equation \(\frac{1}{\cos \theta} - \frac{1}{\cot \theta} = \frac{1}{P}\), we will follow these steps:

### Step 1: Rewrite the terms
We know that \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). Therefore, we can rewrite \(\frac{1}{\cot \theta}\) as:
\[ \frac{1}{\cot \theta} = \frac{\sin \theta}{\cos \theta} \]

### Step 2: Substitute into the equation
Now substitute this back into the equation:
\[ \frac{1}{\cos \theta} - \frac{\sin \theta}{\cos \theta} = \frac{1}{P} \]

### Step 3: Combine the fractions
We can combine the left-hand side:
\[ \frac{1 - \sin \theta}{\cos \theta} = \frac{1}{P} \]

### Step 4: Cross-multiply
Cross-multiplying gives us:
\[ P(1 - \sin \theta) = \cos \theta \]

### Step 5: Use the Pythagorean identity
Recall that \(\cos^2 \theta + \sin^2 \theta = 1\). We can express \(\cos \theta\) in terms of \(\sin \theta\):
\[ \cos \theta = \sqrt{1 - \sin^2 \theta} \]

### Step 6: Substitute \(\cos \theta\) into the equation
Substituting \(\cos \theta\) into the equation from Step 4:
\[ P(1 - \sin \theta) = \sqrt{1 - \sin^2 \theta} \]

### Step 7: Square both sides
To eliminate the square root, we square both sides:
\[ [P(1 - \sin \theta)]^2 = 1 - \sin^2 \theta \]

### Step 8: Expand and rearrange
Expanding the left side gives:
\[ P^2(1 - 2\sin \theta + \sin^2 \theta) = 1 - \sin^2 \theta \]
Rearranging terms leads to:
\[ P^2 - 2P^2\sin \theta + P^2\sin^2 \theta + \sin^2 \theta - 1 = 0 \]

### Step 9: Combine like terms
Combine like terms to form a quadratic equation in terms of \(\sin \theta\):
\[ (P^2 + 1)\sin^2 \theta - 2P^2\sin \theta + (P^2 - 1) = 0 \]

### Step 10: Solve the quadratic equation
Using the quadratic formula \(\sin \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = P^2 + 1\), \(b = -2P^2\), and \(c = P^2 - 1\):
\[ \sin \theta = \frac{2P^2 \pm \sqrt{(-2P^2)^2 - 4(P^2 + 1)(P^2 - 1)}}{2(P^2 + 1)} \]

### Step 11: Find \(\cos \theta\)
After finding \(\sin \theta\), we can use the identity \(\cos^2 \theta = 1 - \sin^2 \theta\) to find \(\cos \theta\).

### Final Result
The value of \(\cos \theta\) can be expressed as:
\[ \cos \theta = \frac{2P}{P^2 + 1} \]