Confusing sines and cosines

If the answer is of the form $\frac{a}{b}$ then one should surely be convinced to simplify the expression using clever manipulations. First we use half-angle cosine formula to transform the square to a linear function.

Let the given expression be denoted by $P$ .

Using half-angle cosine formula we have,

$P=\displaystyle\frac{\cos110^\circ+1}{2}+\displaystyle\frac{2\times\cos55^\circ\sin55^\circ}{2}\sin10^\circ+\sin20^\circ\displaystyle\frac{\cos10^\circ+1}{2}$

$\Rightarrow$ $P=\displaystyle\frac{-\sin20^\circ+1}{2}+\displaystyle\frac{\sin110^\circ}{2}\sin10^\circ+\sin20^\circ\displaystyle\frac{\cos10^\circ+1}{2}$

$\Rightarrow$ $P=\displaystyle\frac{-\sin20^\circ}{2}+\displaystyle\frac{1}{2}+\displaystyle\frac{\cos20^\circ\sin10^\circ}{2}+\displaystyle\frac{\sin20^\circ\cos10^\circ}{2}+\displaystyle\frac{\sin20^\circ}{2}$

$\Rightarrow$ $P=\displaystyle\frac{1}{2}+\displaystyle\frac{\sin10^\circ\cos20^\circ+\sin20^\circ\cos10^\circ}{2}$

$\Rightarrow$ $P=\displaystyle\frac{1}{2}+\displaystyle\frac{\sin30^\circ}{2}$

$\Rightarrow$ $P=\displaystyle\frac{1}{2}+\displaystyle\frac{1}{2\times2}$

$\Rightarrow$ $P=\displaystyle\frac{1}{2}+\displaystyle\frac{1}{4}$

$\Rightarrow$ $P=\displaystyle\frac{3}{4}=\displaystyle\frac{a}{b}$

So $a+b=3+4=\boxed{7}$ .