solution: From the angle addition formula, the numerator is
  \begin{align*}
  &(\cos 5^\circ \cos 20^\circ - \sin 5^\circ \sin 20^\circ) + (\cos 35^\circ \cos 50^\circ -  \sin 35^\circ \sin 50^\circ) \\
  &= \cos (5^\circ + 20^\circ) + \cos (35^\circ + 50^\circ) \\
  &= \cos 25^\circ + \cos 85^\circ.
  \end{align*}From the sum-to-product formula, $\cos 25^\circ + \cos 85^\circ = 2 \cos 55^\circ \cos 30^\circ.$
  
  Similarly, the denominator is
  \begin{align*}
  &\sin 5^\circ \cos 20^\circ - \sin 35^\circ \cos 50^\circ + \cos 5^\circ \sin 20^\circ - \cos 35^\circ \sin 50^\circ) \\
  &= (\sin 5^\circ \cos 20^\circ +  \cos 5^\circ \sin 20^\circ) - (\sin 35^\circ \cos 50^\circ + \cos 35^\circ \sin 50^\circ) \\
  &= \sin (5^\circ + 20^\circ) - \sin (35^\circ + 50^\circ) \\
  &= \sin 25^\circ - \sin 85^\circ \\
  &= -2 \sin 30^\circ \cos 55^\circ,
  \end{align*}so the expression is equal to
  \[\frac{2 \cos 55^\circ \cos 30^\circ}{-2 \sin 30^\circ \cos 55^\circ} = -\frac{\cos 30^\circ}{\sin 30^\circ} = -\sqrt{3} = \tan 120^\circ.\]Hence, the smallest such $\theta$ is $\boxed{120^\circ}.$
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  Find the degree measure of the least positive angle $\theta$ for which
  \[\tan \theta = \frac{\cos 5^\circ \cos 20^\circ + \cos 35^\circ \cos 50^\circ - \sin 5^\circ \sin 20^\circ - \sin 35^\circ \sin 50^\circ}{\sin 5^\circ \cos 20^\circ - \sin 35^\circ \cos 50^\circ + \cos 5^\circ \sin 20^\circ - \cos 35^\circ \sin 50^\circ}.\]
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 120^\circ
extra_info.question: Find the degree measure of the least positive angle $\theta$ for which
  \[\tan \theta = \frac{\cos 5^\circ \cos 20^\circ + \cos 35^\circ \cos 50^\circ - \sin 5^\circ \sin 20^\circ - \sin 35^\circ \sin 50^\circ}{\sin 5^\circ \cos 20^\circ - \sin 35^\circ \cos 50^\circ + \cos 5^\circ \sin 20^\circ - \cos 35^\circ \sin 50^\circ}.\]
extra_info.split: test