Question

Prove that:cos^4 (pi/8)+cos^4 (3pi/8)cos^4 (5pi/8)cos^4 (7pi/8)=sin^4 (pi/8)+sin^4 (3pi/8)sin^4 (5pi/8)sin^4 (7pi/8)=3/2

Answer 1

We have to prove that , cos⁴ (pi/8) + cos⁴(3pi/8) + cos⁴(5pi/8) + cos⁴(7pi/8) = sin⁴(pi/8) + sin⁴(3pi/8) + sin⁴(5pi/8) + sin⁴(7pi/8) = 3/2

we know, cos(pi/8) = cos(pi - 7pi/8) = -cos(7pi/8)

Cos(3pi/8) = cos(pi - 5pi/8) = - cos(5pi/8)

So, cos⁴(7pi/8) + cos⁴(3pi/8) + cos⁴(3pi/8) + cos⁴(7pi/8)

= 2[ cos⁴(7pi/8) + cos⁴(3pi/8) ]

= 2[cos⁴(pi/2 + 3pi/8) + cos⁴(3pi/8) ]

= 2[sin⁴(3pi/8) + cos⁴(3pi/8) ]

= 2[ (sin²(3pi/8) + cos²(3pi/8) )² - 2sin²(3pi/8).cos²(3pi/8) ]

= 2[1 - 2/4[sin²(3pi/4)]

= 2[1 - 1/2{1/√2}²]

= 2 [ 1 - 1/4]

= 3/2 = RHS

Similarly you can prove that, sin⁴(pi/8) + sin⁴(3pi/8) + sin⁴(5pi/8) + sin⁴(7pi/8) = 3/2