`(sec 2A+1 ) sec ^(2) A=`

Options

A. `sec A `
B. `2 sec A`
C. `sec 2A`
D. `2 sec 2A`

Answer

To solve the expression \((\sec 2A + 1) \sec^2 A\), we will follow these steps:

### Step 1: Rewrite in terms of sine and cosine
We know that \(\sec \theta = \frac{1}{\cos \theta}\). Therefore, we can rewrite the expression as:
\[ (\sec 2A + 1) \sec^2 A = \left(\frac{1}{\cos 2A} + 1\right) \cdot \frac{1}{\cos^2 A} \]

### Step 2: Combine the terms inside the parentheses
Now we can combine the terms inside the parentheses:
\[ \frac{1}{\cos 2A} + 1 = \frac{1 + \cos 2A}{\cos 2A} \]
So, the expression becomes:
\[ \frac{1 + \cos 2A}{\cos 2A} \cdot \frac{1}{\cos^2 A} \]

### Step 3: Simplify the expression
Now we can simplify the expression:
\[ \frac{1 + \cos 2A}{\cos 2A \cdot \cos^2 A} \]

### Step 4: Use the identity for \(1 + \cos 2A\)
We can use the trigonometric identity \(1 + \cos 2A = 2 \cos^2 A\):
\[ \frac{2 \cos^2 A}{\cos 2A \cdot \cos^2 A} \]

### Step 5: Cancel out \(\cos^2 A\)
Now we can cancel \(\cos^2 A\) from the numerator and the denominator:
\[ \frac{2}{\cos 2A} \]

### Step 6: Rewrite in terms of secant
Finally, we can rewrite this in terms of secant:
\[ 2 \sec 2A \]

Thus, the final answer is:
\[ 2 \sec 2A \]

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