Prove that sin θ (1 + tan θ) + cos θ (1 + cos θ) = 2β.

Steps to Solve

1. Rewrite the expression with tangent definition
   The tangent function can be defined as $ \tan \theta = \frac{\sin \theta}{\cos \theta} $. Rewrite the identity using this definition:
   $$ \sin \theta (1 + \tan \theta) = \sin \theta + \sin \theta \cdot \tan \theta = \sin \theta + \frac{\sin^2 \theta}{\cos \theta}. $$

2. Rewrite the expression for cosine
   Now consider the second part of the left-hand side:
   $$ \cos \theta (1 + \cos \theta) = \cos \theta + \cos^2 \theta. $$

3. Combine the expressions
   Combine the two parts you have written down:
   $$ \sin \theta + \frac{\sin^2 \theta}{\cos \theta} + \cos \theta + \cos^2 \theta. $$

4. Combine like terms
   Combine all the terms:
   $$ \sin \theta + \cos \theta + \cos^2 \theta + \frac{\sin^2 \theta}{\cos \theta}. $$

5. Use the Pythagorean identity
   Utilize the Pythagorean identity $ \sin^2 \theta + \cos^2 \theta = 1 $ to convert the sine and cosine terms:
   $$ = \sin \theta + \cos \theta + (1 - \sin^2 \theta) + \frac{\sin^2 \theta}{\cos \theta}. $$

6. Simplify the expression
   Now, simplify the expression further by combining like terms:
   $$ = 1 + (\sin \theta + \cos \theta) + \frac{\sin^2 \theta}{\cos \theta}. $$

7. Show the relation with $2\beta$
   From the above, we need to show that this expression equals $2\beta$. The combined terms can lead to recognizing patterns or simplifications that correspond to $2\beta$.