सिद्ध कीजिए कि -
` cos^(4). pi/8 + cos^(4) (3pi)/8 + cos^(4). (5pi)/8 + cos^(4). (7pi)/8 = 3/2 `

Explanation

` L.H.S = cos ^(4). pi/8 + cos^(4) (3pi)/8 + cos^(4). (5pi)/8 +cos^(4).(7pi)/8`
` = cos^(4). pi/8 + cos^(4). (3pi)/8 + [ cos ( pi- (3pi)/8)]^(4)`
` + [ cos (pi-pi/8)]^(4)`
` = cos^(4) . pi/8 + cos^(4) (3pi)/8 + [ -cos.(3pi)/8]^(4)`
` + [ -cos . pi/8]^(4)`
` = 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]`
` = 1/2 [ (2 cos^(2) . pi/2 )^(2) + ( 2 cos^(2). (3pi)/8)^(2)]`
` = 1/2 [ ( 1+ cos .pi/4)^(2) + ( 1+ cos. (3pi)/4)^(2)]`
` = 1/2 [ (1+ 1/(sqrt(2)))^(2) + { 1+ cos (pi - pi/4)}^(2)]`
` = 1/2 [(1+ 1/(sqrt(2)))^(2)+(1+cos.(3pi)/4)^(4)]`
` = 1/2 [ (1+cos.pi/4)^(2)+ (1+cos.(3pi)/4)^(2)]`
` ( :' 2 cos^(2) theta =1 + cos 2 theta )`
` = 1/2 [ ( 1+ 1/(sqrt(2)))^(2)+ {1 + cos (pi - pi/4)}^(2)]`
`= 1/2 [ (1+1/(sqrt(2)))^(2) + (1- cos.pi/4)^(2)]`
` = 1/2 [ ( 1+1/(sqrt(2)))^(2) + (1- 1/(sqrt(2)))^(2)]`
` = 1/2 [ 1 + 1/2 + 2/(sqrt(2))+1 + 1/2 - 2/(sqrt(2))]`
` = 3/2 = R.H.S`