`(sinalpha.sin3alpha + sin3alpha.sin7alpha + sin5alpha.sin15alpha)/(sinalpha.cos3alpha +sin3alphacos7alpha +sin5alpha.cos15alpha)=`

(sin alpha*sin3 alpha+sin3 alpha*sin7 alpha+sin5 alpha*sin15 alpha)/(sin alpha*cos3 alpha+sin3 alpha cos7 alpha+sin5 alpha*cos15 alpha)=

( sin alpha sin 3alpha + sin 3alpha sin 7 alpha + sin 5 alpha sin 15 alpha )/( sin alpha cos 3 alpha + sin 3alpha cos 7 alpha + sin 5 alpha cos 15 alpha ) =

(sin 5 alpha-sin3alpha)/(cos 5 alpha+2 cos 4 alpha+cos 3alpha)=

sin alpha+sin2 alpha+sin4 alpha+sin5 alpha=4cos((alpha)/(2))cos(3(alpha)/(2))sin3 alpha

sin alpha+sin2 alpha+sin4 alpha+sin5 alpha=4cos((alpha)/(2))cos(3(alpha)/(2))sin3 alpha

tan theta = (sin alpha-cos alpha) / (sin alpha + cos alpha)

Prove the following identities: (sin alpha sin 11 alpha + sin 3 alpha sin 7 alpha)/(sin alpha cos 11 alpha + sin 3 alpha cos 7 alpha)= tan 8 alpha

prove that (sin alpha+sin3 alpha+sin5 alpha)/(cos alpha+cos3 alpha+cos5 alpha)=tan3 alpha

((cos alpha-cos3 alpha)(sin8 alpha+sin2 alpha))/((sin5 alpha-sin alpha)(cos4 alpha-cos6 alpha))=