Using high school geometry only, show
that sin(A+B) = sinAcosB + sinBcosA
how would i show how one side equals the other side??
consider
in this figure, we have that NOPR is a rectangle, ON was extended and we get the point M. Join M and P and extend NR, point Q borns. Finally make QP orthogonal to MP and join M and Q.
let $\displaystyle \measuredangle\,OMP=\alpha$ and $\displaystyle \measuredangle\,QMP=\beta.$ It's easy to prove that $\displaystyle \measuredangle\,OMP=\measuredangle\,PQR=\alpha.$
compute $\displaystyle \sin(\alpha+\beta)$ and we have it's equal to $\displaystyle \frac{{\overline {QN} }}{{\overline {QM} }} = \frac{{\overline {QR} + \overline {RN} }}{{\overline {QM} }} = \frac{{\overline {QR} }}{{\overline {QM} }} + \frac{{\overline {OP} }}{{\overline {QM} }}.$ Multiply the first quotient top and bottom by $\displaystyle \overline{QP}$ and in the same fashion for the second quotient by $\displaystyle \overline{MP}$ and you'll get the identity.