Question

Show that Sin^4 π/8 + Sin^4 3π/8 + Sin^4 5π/8 + Sin^4 7π/8 = 3/2.

Answer 1

Solution - 
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

Answer 2

Answer:

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