Question

Write the expression in terms of sin x and cos x only. sin(2x)+cos(3x)

Answer 1

Sin2x=2sinxcosx, cos2x=1-2sin^2x
sin(2x)+cos(3x)=2sinxcosx+cos(x+2x)
cos(x+2x)=cosx(1-2sin^2(x))-sinx2sinxcosx
sin(2x)+cos(3x)=2sinxcosx(1-sinx)+cosx(1-2sin^2(x))

Answer 2

Use the following identities:
$\sin{2x}=2\sin{x}\cos{x}\\ \cos{2x}=\cos^2{x}-\sin^2{x} \\ \cos{(x+y)}=\cos{x}\cos{y}-\sin{x}\sin{y}$

$\sin{2x}+\cos{3x} \\=2\sin{x}\cos{x}+\cos{(2x+x)} \\=2\sin{x}\cos{x}+\cos{2x}\cos{x}-\sin{2x}\sin{x} \\=2\sin{x}\cos{x}+(\cos^2{x}-\sin^2{x})\cos{x}-2\sin{x}\cos{x}\sin{x} \\=2\sin{x}\cos{x}+\cos^3{x}-\sin^2{x}\cos{x}-2\sin^2{x}\cos{x} \\=2\sin{x}\cos{x}+\cos^3{x}-3\sin^2{x}\cos{x} \\=\sin{x}\cos{x}(2+\cos^2{x}-3\sin{x})$