Trigonometry

(a) Show that Sin^2 nx－Sin^2 mx = Sin(n+m)x
Sin(n－m)x

Sol

Sin(n+m)x Sin(n－m)x

=(SinnxCosmx+CosnxSinmx)(SinnxCosmx－CosnxSinmx)

=Sin^2 nxCos^2 mx－Cos^2 nxSin^2 mx

=Sin^2 nx(1－Sin^2 mx)－(1－Sin^2 nx)Sin^2 mx

=Sin^2 nx－Sin^2 nxSin^2 mx－Sin^2
mx+Sin^2 nxSin^2 mx

=Sin^2 nx－Sin^2 mx



(b) Hence，solve the equation for Sin^2 3x－Sin^2 2x－Sinx = 0

for 0 <=x<=π

Sol

Sin^2 nx－Sin^2 mx = Sin(n+m)x Sin(n－m)x

So

Sin^2 3x－Sin^2 2x=Sin5xSinx

Sin^2 3x－Sin^2 2x－Sinx = 0

Sin5xSinx－Sinx=0

Sinx(Sin5x－1)=0

Sinx=0 or Sin5x=1

0 <=x<=π

(1) Sinx=0

x=0 or x=π

(2) 0<=5x<=5π

Sin5x=1

5x=π/2 or 5x=5π/2 or 5x=9π/2

x=π/10 or x=π/2
or x=9π/10