यदि `sin theta+sin phi =sqrt(3) ( cos phi -cos theta ) ` साबित करें कि `sin 3theta + sin 3 phi = 0 `

Explanation

प्रश्न से, `2sin. ( theta + phi )/(2) cos. ( theta - phi)/(2)= sqrt(3).2sin. (phi+theta)/(2) sin. (theta-phi)/(2)`
या `sin. (theta+phi)/(2) [ cos .(theta-phi)/(2)-sqrt(3)sin. (theta-phi)/(2)] = 0 `
`:. sin . (theta +phi)/(2) =0` या `cos . ( theta - phi)/(2)-sqrt(3) sin . (theta - phi)/(2) = 0`
`:. (theta+phi)/(2) = 0^(@)` या `tan .(theta-phi)/(2) = (1)/(sqrt(3))` अर्थात `(theta-phi)/(2) = 30^(@)`
अब `sin 3theta + sin 3 phi = 2 sin. (3( theta+ phi))/(2) cos. (3(theta-phi))/(2) = 0 `
`[ :'` जब `( theta+ phi)/(2) = 0^(@)" " sin . (3(theta+phi))/(2) = 0` तथा जब `( theta- phi)/(2) = 30^(@) " " cos. (3( theta -phi))/(2) = cos 90^(@) = 0`]