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

Explanation

To prove the equation \((\tan \theta + 2)(2 \tan \theta + 1) = 5 \tan \theta + 2 \sec^2 \theta\), we will start with the left-hand side (LHS) and manipulate it to show that it equals the right-hand side (RHS).

### Step-by-Step Solution:

1. **Start with the Left-Hand Side (LHS)**:
\[ \text{LHS} = (\tan \theta + 2)(2 \tan \theta + 1) \]
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