How to simplify trigonometric expression I have some trouble with my homework. The problem is to simplify the following trigonometric expression:
$$
\frac{\sin150^{\circ}-\cos240^{\circ}}{\cot730^{\circ}\cot800^{\circ}+\tan730^{\circ}\tan800^{\circ}}
$$ 
I know that $\sin{150^{\circ}} = \sin{\frac{5\pi}{3}} = \frac{1}{2}$ and $\cos240^{\circ} = \cos{\frac{4\pi}{3}}=-\frac{1}{2}$, so $\sin150^{\circ}-\cos240^{\circ} = 1$, and numerator equals $1$. How do I deal with denominator?
 A: Due to periodic properties of sine and cosine we have $\cot730^{\circ} = \frac{1}{\tan730^{\circ}} = \cot10^{\circ} = \frac{1}{\tan10^{\circ}}$ and $\cot800^{\circ} = \frac{1}{\tan800^{\circ}} = \cot80^{\circ} = \frac{1}{\tan80^{\circ}}$. Therefore:
$$
\frac{1}{\cot730^{\circ}\cot800^{\circ}+\tan730^{\circ}\tan800^{\circ}}
=
\frac{1}{\cot10^{\circ}\cot80^{\circ}+\tan10^{\circ}\tan80^{\circ}}
$$ 
Now note that 
$$\cos10^{\circ} = \cos(-10^{\circ}) = \sin(-10^{\circ} + 90^{\circ}) = \sin80^{\circ},$$ 
$$\cos80^{\circ} = -\sin(80^{\circ}-90^{\circ})= -\sin(-10^{\circ}) = \sin10^{\circ},$$ 
so
$$
\tan10^{\circ} = \frac{\sin10^{\circ}}{\cos10^{\circ}}= \frac{\cos80^{\circ}}{\sin80^{\circ}} = \cot80^{\circ},
$$
$$
\tan80^{\circ} = \frac{\sin80^{\circ}}{\cos80^{\circ}}= \frac{\cos10^{\circ}}{\sin10^{\circ}} = \cot10^{\circ},
$$
so you have
$$
\frac{1}{\cot10^{\circ}\cot80^{\circ}+\tan10^{\circ}\tan80^{\circ}}
=
\frac{1}{\cot10^{\circ}\tan10^{\circ}+\tan10^{\circ}\cot10^{\circ}}
=
\frac{1}{2}
$$ 
A: Let me give you some hints:


*

*$\cot A = \tan (90^{\circ} - A)$ and vice-versa 

*and $\tan A=\frac{1}{\cot A}$ and vice-versa

*$\cot730^{\circ}=\cot10^{\circ} $

*$\cot800^{\circ}=\cot80^{\circ}$
Try to figure out the rest by yourself.
