साबित करें कि
`cos^(4) . ( pi)/(8) + cos^(4) . ( 3pi)/( 8) + cos^(4) .( 5pi)/(8) + cos^(4) . ( 7pi)/( 8) = ( 3)/( 2)`

Explanation

`cos^(4) . ( pi)/(8)= ( cos^(2) . ( pi)/( 8) )^(2) = ((1+ cos. ( pi)/( 4))/(2))^(2) = ((1+ (1)/(sqrt(2)))/(2))^(2) = (sqrt(2 +1)/(2sqrt(2)))^(2)`
`= ( 2+1+ 2 sqrt(2))/( 8) = ( 3)/( 8) + ( sqrt(2))/( 4)`
`cos^(4). ( 3pi)/(8) = ( cos^(2) . ( 3pi)/(8))^(2) = ((1+ cos. ( 3pi)/(4))/(2))^(2)`
` = (1- (1)/(sqrt(2))/(2))^(2) = (( sqrt(2)-1)/(2sqrt(2)))^(2) = ( 2+1-2sqrt(2))/(8) = (3)/(8) - ( sqrt(2))/(4)`
`cos. ( 5pi)/(8) = cos (pi- ( 3pi)/(8)) = - cos .( 3pi)/(8)`
`:. cos^(4) . ( 5pi)/(8) = cos^(4) . ( 3pi)/( 8)`
पुनः `cos. ( 7pi)/(8) = cos (pi- ( pi)/(8)) = - cos. ( pi)/(8)`
`:. cos^(4) . ( 7pi)/( 8) = cos^(4) . ( pi)/( 8)`
अब बायाँ पक्ष ` cos^(4) . ( pi)/(4) + cos^(4) . ( 3pi)/( 8) + cos^(4). ( 5pi)/( 8) + cos^(4). (7pi)/(8)`
` cos^(4). ( pi)/( 8) + cos^(4). ( 3pi)/( 8) + cos^(4). ( 3pi)/( 8) + cos^(4) . ( pi)/( 8) `
`= 2 ( cos^(4). ( pi)/( 8) + cos^(4). ( 3pi)/( 8) )`
`= 2 ((3)/( 8) + ( sqrt(2))/( 4) + ( 3)/( 8) - ( sqrt(2))/(4)) = 2 xx ( 3)/(4) = (3)/(2)`