I'm preparing for a math competition, and was stumped by this problem. The original problem is shown below, and the correct answer is $120°$. I'm posting this here to ask for explanation on how this answer was reached.
The equation given is $$\tan\theta={{\cos5°\cos20°+\cos35°\cos50°-\sin5°\sin20°-\sin35°\sin50°}\over{\sin5°\cos20°-\sin35°\cos50°+\cos5°\sin20°-\cos35°\sin50°}}$$
I was able to simplify the numerator using $\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)$, and simplify the denominator using $\sin(x+y)=\sin(x)\cos(y)+\sin(y)\cos(x)$ to get the expression given in the title of this question, but I don't know how to go any further, or if this is even the right first step.
Best Answer
We have: $\tan(\theta)=\dfrac{\cos(25^{\circ})+\cos(85^{\circ})}{\sin(25^{\circ}) - \sin(85^{\circ})}=\dfrac{2\cos(55^{\circ})\cos(30^{\circ})}{-2\cos(55^{\circ})\sin(30^{\circ})}=-\cot(30^{\circ})=-\sqrt{3} \implies \theta = 120^{\circ}$ would be the smallest positive angles satisfy the equation.