`costheta(tantheta+2)(2tantheta+1)=?`

Options

A. 0
B. 1
C. `2sectheta+5sintheta`
D. `5sectheta+2sintheta`

Answer

To solve the expression \( \cos \theta (\tan \theta + 2)(2 \tan \theta + 1) \), we can follow these steps:

### Step 1: Expand the expression
We start by expanding the expression inside the parentheses:
\[ (\tan \theta + 2)(2 \tan \theta + 1) \]
Using the distributive property (also known as the FOIL method for two binomials):
\[ = \tan \theta \cdot 2 \tan \theta + \tan \theta \cdot 1 + 2 \cdot 2 \tan \theta + 2 \cdot 1 \]
\[ = 2 \tan^2 \theta + \tan \theta + 4 \tan \theta + 2 \]
\[ = 2 \tan^2 \theta + 5 \tan \theta + 2 \]

### Step 2: Multiply by \( \cos \theta \)
Now we multiply the expanded expression by \( \cos \theta \):
\[ \cos \theta (2 \tan^2 \theta + 5 \tan \theta + 2) \]
Distributing \( \cos \theta \):
\[ = 2 \cos \theta \tan^2 \theta + 5 \cos \theta \tan \theta + 2 \cos \theta \]

### Step 3: Substitute \( \tan \theta \)
Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Therefore, \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \):
\[ = 2 \cos \theta \left(\frac{\sin^2 \theta}{\cos^2 \theta}\right) + 5 \cos \theta \left(\frac{\sin \theta}{\cos \theta}\right) + 2 \cos \theta \]
\[ = \frac{2 \sin^2 \theta}{\cos \theta} + 5 \sin \theta + 2 \cos \theta \]

### Step 4: Combine terms
The expression can be simplified further if necessary, but we can leave it in this form:
\[ = \frac{2 \sin^2 \theta + 5 \sin \theta \cos \theta + 2 \cos^2 \theta}{\cos \theta} \]

### Final Result
Thus, the final expression is:
\[ \frac{2 \sin^2 \theta + 5 \sin \theta \cos \theta + 2 \cos^2 \theta}{\cos \theta} \]