`sin5theta = sin(3theta+2theta)`
`=sin3thetacos2theta +cos3thetasin2theta`
`=(3sintheta-4sin^3theta)(1-2sin^2theta) + (4cos^3theta-3costheta)(2sinthetacostheta)`
`=(3sintheta-4sin^3theta)(1-2sin^2theta) + (4cos^2theta-3)(2sinthetacos^2theta)`
`=(3sintheta-4sin^3theta)(1-2sin^2theta) + (4(1-sin^2theta)-3)(2sintheta(1-sin^2theta))`
`=(3sintheta-4sin^3theta)(1-2sin^2theta) + (4-4sin^2theta-3)(2sintheta-2sin^3theta))`
`=(3sintheta-4sin^3theta)(1-2sin^2theta) + (1-4sin^2theta)(2sintheta-2sin^3theta))`
`=3sintheta-4sin^3theta-6sin^3theta+8sin^5theta+2sintheta-8sin^3theta-2sin^3theta+8sin^5theta`
`=16sin^5theta-20sin^3theta+5sintheta`
`:. sin5theta = 16sin^5theta-20sin^3theta+5sintheta.`
