cos⁸A+sin⁸A = (Cos⁴A)² + (Sin⁴A)²
=> (Cos⁴A + Sin⁴A)² - 2Cos⁴ASin⁴A
=> [(Cos²A)²+(Sin²A)²]² - 2(CosASinA)⁴
=> [(Cos²A+Sin²A)² - 2Cos²ASin²A]² - 2(CosASinA)⁴
=> [1 - 2Cos²ASin²A]² - 2(CosASinA)⁴
=> [1 - 2(CosASinA)²]² - 2(CosASinA)⁴
//Remember: Sin2A = 2SinA CosA => SinACosA = 1/2Sin2A
=> [1 - 2(1/2Sin2A)²]²- 2(1/2Sin2A)⁴
=> [1 - 1/2Sin²2A]² - 1/8Sin⁴2A
=> 1 + 1/4Sin⁴2A - Sin²2A - 1/8Sin⁴2A
=> 1 - Sin²2A + 1/8Sin⁴2A
=> R.H.S
