Hello, desperate_on_sunday_night!

ok the answer 1 apparently because i went on hotmath.com and it explained it like this

$\displaystyle \cos35\sin55 + \cos55\sin35$

$\displaystyle \sin^2\!55 + \cos^2\!35 \:=\:1$

ok i understand that . . . . you do? If that's all they wrote, very sloppy!!

We're expected to visualize this right triangle . . . Code:

*
* *
c * 55° *
* * a
* *
* 35° *
* * * * * * *
b

$\displaystyle \begin{array}{cccccc}\text{We see that:} &\cos35 & = & \frac{b}{c} &=& \sin 55^o \\

\text{and that:} & \sin35 &=& \frac{a}{c} &=& \cos55^o\end{array}$

. We have: .$\displaystyle \underbrace{\cos35}_{\downarrow}\sin55 + \cos55\underbrace{\sin35}_{\downarrow}$

Substitute: .$\displaystyle \overbrace{\sin55}\sin55 + \cos55\overbrace{\cos55} \;\;=\;\;\sin^2\!55 + \cos^2\!55 \;\;=\;\;1$