`2sin^(2) theta + 4 cos (theta + alpha) sin alpha sin theta +cos 2 ( alpha+ theta)`
`= 2 sin^(2) theta + 2 cos ( theta + alpha) 2sinalphasin theta + cos 2 ( alpha + theta)`
`= 2 sin^(2) theta+ 2 cos (theta + alpha) [cos ( theta – alpha) -cos ( theta + alpha) ] + cos 2 ( alpha+ theta)`
`= 2 sin^(2) theta+ 2 cos (theta + alpha) cos ( theta – alpha) -2cos^(2) ( theta + alpha) + cos 2 ( alpha+ theta)`
`=2 sin^(2) theta+2(cos^(2) theta-sin^(2) alpha) – 2 cos^(2) ( theta + alpha) + [2 cos^(2) ( alpha+ theta) -1]`
`= 2 sin^(2) theta + 2 cos^(2) theta – 2sin^(2) alpha -1 `
`= 2( sin ^(2) theta + cos^(2) theta ) – 2 sin^(2) alpha-1 `
`= 2-2-sin^(2) alpha-1= 1- 2 sin^(2) alpha = cos2 alpha`
जो कि `theta ` से स्वतंत्र है ।