`(tantheta+2)(2 tantheta+1) = 5 tantheta+ sec^(2)theta`

Options

A. `5 tantheta+ sectheta`
B. `5 tantheta+ 2sec^(2)theta`
C. `5 tan^(2)theta+ sec^(2)theta`
D. none of these

Explanation

To solve the equation \((\tan \theta + 2)(2 \tan \theta + 1) = 5 \tan \theta + \sec^2 \theta\), we will follow these steps:

### Step 1: Expand the left-hand side
We start by expanding the left-hand side of the equation:
\[ (\tan \theta + 2)(2 \tan \theta + 1) \]
Using the distributive property (FOIL method):
\[ = \tan \theta \cdot 2 \tan \theta + \tan \theta \cdot 1 + 2 \cdot 2 \tan \theta + 2 \cdot 1 \]
\[ = 2 \tan^2 \theta + \tan \theta + 4 \tan \theta + 2 \]
\[ = 2 \tan^2 \theta + 5 \tan \theta + 2 \]

### Step 2: Rewrite the right-hand side
The right-hand side of the equation is:
\[ 5 \tan \theta + \sec^2 \theta \]

### Step 3: Use the identity for \(\sec^2 \theta\)
We know from trigonometric identities that:
\[ \sec^2 \theta = 1 + \tan^2 \theta \]
Substituting this into the right-hand side gives:
\[ 5 \tan \theta + (1 + \tan^2 \theta) = 5 \tan \theta + 1 + \tan^2 \theta \]

### Step 4: Set the two sides equal
Now we set the expanded left-hand side equal to the modified right-hand side:
\[ 2 \tan^2 \theta + 5 \tan \theta + 2 = 5 \tan \theta + 1 + \tan^2 \theta \]

### Step 5: Simplify the equation
Subtract \(5 \tan \theta\) from both sides:
\[ 2 \tan^2 \theta + 2 = 1 + \tan^2 \theta \]
Now, subtract \(\tan^2 \theta\) from both sides:
\[ 2 \tan^2 \theta - \tan^2 \theta + 2 = 1 \]
This simplifies to:
\[ \tan^2 \theta + 2 = 1 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ \tan^2 \theta = 1 - 2 \]
\[ \tan^2 \theta = -1 \]

### Step 7: Analyze the result
Since \(\tan^2 \theta\) cannot be negative, there are no real solutions to this equation.

### Final Conclusion
Thus, the equation \((\tan \theta + 2)(2 \tan \theta + 1) = 5 \tan \theta + \sec^2 \theta\) has no real solutions.

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