\operatorname { sin } ^ { 4 } \frac { \pi } { 8 } + \operatorname { si

1.	\operatorname { sin } ^ { 4 } \frac { \pi } { 8 } + \operatorname { sin } ^ { 4 } \frac { 3 \pi } { 8 } + \operatorname { sin } ^ { 4 } \frac { 5 \pi } { 8 } + \operatorname { sin } ^ { 4 } \frac { 7 \pi } { 8 } = \frac { 3 } { 2 }
Answer» sin⁴ (π / 8) + sin⁴ ( 3π/8) + sin⁴( 5π/8 ) + sin⁴( 7π/8 ) = [sin²(π/8)]² + [sin²( 3π/8 )]² + [sin²( 5π/8 )]² + [sin²( 7π/8 )]² = [{1 - cos(π/4)}/2]² + [{1 - cos(3π/4)}/2]² + [{1 - cos(5π/4)}/2]² + [{1 - cos(7π/4)}/2]² = [{1 - √2/2 }/2]² + [{1 + √2/2 }/2]² + [{1 + √2/2}/2]² + [{1 - √2/2}/2]² = 2[{1 - √2/2 }/2]² + 2[{1 + √2/2 }/2]² = 2[{2 - √2}/4]² + 2[{2 + √2}/4]² = (1/8)(2 - √2)² + (1/8)(2 + √2)² = (1/8)[(4 - 4√2 + 2) + (4 + 4√2 + 2)] = 12/8 = 3/2

\operatorname { sin } ^ { 4 } \frac { \pi } { 8 } + \operatorname { sin } ^ { 4 } \frac { 3 \pi } { 8 } + \operatorname { sin } ^ { 4 } \frac { 5 \pi } { 8 } + \operatorname { sin } ^ { 4 } \frac { 7 \pi } { 8 } = \frac { 3 } { 2 }

Answer»

sin⁴ (π / 8) + sin⁴ ( 3π/8) + sin⁴( 5π/8 ) + sin⁴( 7π/8 )

= [sin²(π/8)]² + [sin²( 3π/8 )]² + [sin²( 5π/8 )]² + [sin²( 7π/8 )]²

= [{1 - cos(π/4)}/2]² + [{1 - cos(3π/4)}/2]² + [{1 - cos(5π/4)}/2]² + [{1 - cos(7π/4)}/2]²

= [{1 - √2/2 }/2]² + [{1 + √2/2 }/2]² + [{1 + √2/2}/2]² + [{1 - √2/2}/2]²

= 2[{1 - √2/2 }/2]² + 2[{1 + √2/2 }/2]²

= 2[{2 - √2}/4]² + 2[{2 + √2}/4]²

= (1/8)(2 - √2)² + (1/8)(2 + √2)²

= (1/8)[(4 - 4√2 + 2) + (4 + 4√2 + 2)]

= 12/8 = 3/2