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

Answer

To prove the equation \((\tan \theta + 2)(2\tan \theta + 1) = 5\tan \theta + 2\sec^2 \theta\), we will start by expanding the left-hand side (LHS) and simplifying it step by step.

### Step-by-Step Solution:

1. **Expand the Left-Hand Side:**
\[ (\tan \theta + 2)(2\tan \theta + 1) \]
We will use the distributive property (also known as the FOIL method for 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. **Combine Like Terms:**
Now, we combine the like terms:
\[ = 2\tan^2 \theta + (1 + 4)\tan \theta + 2 \]
\[ = 2\tan^2 \theta + 5\tan \theta + 2 \]

3. **Use the Identity for \(\sec^2 \theta\):**
We know from trigonometric identities that:
\[ \sec^2 \theta = 1 + \tan^2 \theta \]
We can rewrite \(2\sec^2 \theta\) as:
\[ 2\sec^2 \theta = 2(1 + \tan^2 \theta) = 2 + 2\tan^2 \theta \]

4. **Substituting Back:**
Now, substitute \(2\sec^2 \theta\) back into our expression:
\[ = 5\tan \theta + 2 + 2\tan^2 \theta \]

5. **Rearranging:**
We can rearrange this to match the right-hand side (RHS):
\[ = 5\tan \theta + 2 + 2\tan^2 \theta \]
\[ = 5\tan \theta + 2 + 2\tan^2 \theta \]
This shows that both sides are equal.

6. **Conclusion:**
Therefore, we have shown that:
\[ (\tan \theta + 2)(2\tan \theta + 1) = 5\tan \theta + 2\sec^2 \theta \]
Hence, the equation is proved.