If sec `theta` +tan `theta =P` , then cos `theta ` is equal to

Options

A. `(p^1+1)/(p^2 -1)`
B. `(p^2 -1)/(p^2 +1)^2`
C. `(2p)/(p^2 +1)`
D. `(4 p^2)/(p^2 +1)^2`

Answer

To solve the problem, we start with the equation given:

**Step 1: Start with the equation**
\[ \sec \theta + \tan \theta = P \]

**Hint:** Recall the definitions of secant and tangent in terms of cosine and sine.

**Step 2: Use the identity**
We know the identity:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
This can be rearranged to express \(\tan^2 \theta\):
\[ \tan^2 \theta = \sec^2 \theta - 1 \]

**Hint:** This identity is derived from the Pythagorean theorem in trigonometry.

**Step 3: Substitute \(\tan \theta\)**
Taking the square root of both sides gives:
\[ \tan \theta = \sqrt{\sec^2 \theta - 1} \]
Now substitute this into the original equation:
\[ \sec \theta + \sqrt{\sec^2 \theta - 1} = P \]

**Hint:** Make sure to isolate the square root for easier manipulation.

**Step 4: Isolate the square root**
Rearranging gives:
\[ \sqrt{\sec^2 \theta - 1} = P - \sec \theta \]

**Hint:** This will help us square both sides in the next step.

**Step 5: Square both sides**
Squaring both sides results in:
\[ \sec^2 \theta - 1 = (P - \sec \theta)^2 \]

**Hint:** Remember to expand the right side correctly.

**Step 6: Expand the right side**
Expanding gives:
\[ \sec^2 \theta - 1 = P^2 - 2P \sec \theta + \sec^2 \theta \]

**Hint:** Notice that \(\sec^2 \theta\) appears on both sides.

**Step 7: Simplify the equation**
Subtract \(\sec^2 \theta\) from both sides:
\[ -1 = P^2 - 2P \sec \theta \]
Rearranging gives:
\[ 2P \sec \theta = P^2 + 1 \]

**Hint:** This step is crucial for finding \(\sec \theta\).

**Step 8: Solve for \(\sec \theta\)**
Dividing both sides by \(2P\) gives:
\[ \sec \theta = \frac{P^2 + 1}{2P} \]

**Hint:** Remember that \(\sec \theta = \frac{1}{\cos \theta}\).

**Step 9: Find \(\cos \theta\)**
Taking the reciprocal gives:
\[ \cos \theta = \frac{2P}{P^2 + 1} \]

**Final Answer:**
\[ \cos \theta = \frac{2P}{P^2 + 1} \]