Question

solve : cos^4(π/8)+cos^4(3π/8)+cos^4(5π/8)+cos^4(7π/8)=?

Answer 1

cos⁴π/8 + cos⁴3π/8 + cos⁴(π - 3π/8) + cos⁴(π - π/8)

⇒ 2cos⁴π/8 + 2cos⁴3π/8

⇒ 2{cos⁴π/8 + cos⁴(π/2 - π/8)

⇒ 2{ cos⁴π/8 + sin⁴π/8 }

⇒ 2{1 - 2sin²π/8 × cos²π/8}

⇒2{ 1 - sinπ/4 × sinπ/8 × cosπ/8}

⇒ {2 - sinπ/4 × 2sinπ/8 × cosπ/8}

⇒{ 2 - sinπ/4 × sinπ/4}

⇒{ 2- 1/2} = 3/2

identities used to solve the problem are

1. a² + b² = {a + b}² - 2ab - - here a = sin²π/8 and b = cos²π/8
2. 2sinxcosx = sin2x - - here x = π/8