Prove that:\cos ^{4} \pi / 8+\cos ^{4} 3 \pi / 8+\cos ^{4} 5 \pi / 8+\

1.	Prove that:\cos ^{4} \pi / 8+\cos ^{4} 3 \pi / 8+\cos ^{4} 5 \pi / 8+\cos ^{4} 7 \pi / 8=3/2
Answer» Cos⁴π/8+cos⁴3π/8+cos⁴5π/8+cos⁴7π/8 =cos⁴π/8+cos⁴3π/8+{cos(π/2+π/8)}⁴+{cos(π/2+3π/8)}⁴ =cos⁴π/8++cos⁴3π/8+(-sinπ/8)⁴+(-sin3π/8)⁴ =sin⁴π/8+cos⁴π/8+sin⁴3π/8+cos⁴3π/8 ={(sin²π/8)²+(cos²π/8)²}+{(sin²3π/8)²+(cos²3π/8)²} ={(sin²π/8+cos²π/8)²-2sin²π/8cos²π/8}+{(sin²3π/8+cos²3π/8)²- 2sin²3π/8cos²3π/8} =1-2sin²π/8cos²π/8+1-2sin²3π/8cos²3π²/8 =(1/2){4-(2sinπ/8cosπ/8)²-(2sin3π/8cos3π/8)²} =(1/2)[4-(sinπ/4)²-(sin3π/4)²][∵, sin2A=2sinAcosA] =(1/2)[4-(1/√2)²-(cosπ/4)²][∵, sin3π/4=sin{(π/2×1)+π/4}=cosπ/4] =(1/2)[4-1/2-(1/√2)²] =(1/2)[4-(1/2+1/2)] =(1/2)(4-1) =3/2 (Proved)

Prove that:\cos ^{4} \pi / 8+\cos ^{4} 3 \pi / 8+\cos ^{4} 5 \pi / 8+\cos ^{4} 7 \pi / 8=3/2

Answer»

Cos⁴π/8+cos⁴3π/8+cos⁴5π/8+cos⁴7π/8

=cos⁴π/8+cos⁴3π/8+{cos(π/2+π/8)}⁴+{cos(π/2+3π/8)}⁴

=cos⁴π/8++cos⁴3π/8+(-sinπ/8)⁴+(-sin3π/8)⁴

=sin⁴π/8+cos⁴π/8+sin⁴3π/8+cos⁴3π/8

={(sin²π/8)²+(cos²π/8)²}+{(sin²3π/8)²+(cos²3π/8)²}

={(sin²π/8+cos²π/8)²-2sin²π/8cos²π/8}+{(sin²3π/8+cos²3π/8)²- 2sin²3π/8cos²3π/8}

=1-2sin²π/8cos²π/8+1-2sin²3π/8cos²3π²/8

=(1/2){4-(2sinπ/8cosπ/8)²-(2sin3π/8cos3π/8)²}

=(1/2)[4-(sinπ/4)²-(sin3π/4)²][∵, sin2A=2sinAcosA]

=(1/2)[4-(1/√2)²-(cosπ/4)²][∵, sin3π/4=sin{(π/2×1)+π/4}=cosπ/4]

=(1/2)[4-1/2-(1/√2)²]

=(1/2)[4-(1/2+1/2)]

=(1/2)(4-1)

=3/2 (Proved)