Question

The value of sin36∘sin72∘sin108∘sin144∘ is equal to

Answer 1

=> sin 36 * sin 72 * sin 108 * sin 144

=> sin 36 * sin 72 * sin (180 - 72) * sin (180 - 36)

=> sin 36 * sin 72 * sin 72 * sin 36

=> [sin 36 * sin 72]²

=>¼ [ 2sin 36 * sin 72]²

=>¼ [ cos 36 - cos 108]²

2sin(A)*sin(B) = [cos(A-B) - cos(A+B)]

=>¼ [ cos 36 + sin 18]² ; cos(108) = cos(90+18) = -sin(18)]

=>¼ [ (√5 + 1)/4 + (√5 - 1)/4]²

; put

cos 36 = (√5 + 1)/4 and sin 18 = (√5 - 1)/4

=>¼ [ √5/2 ]²

=> 5/16

Answer 2

Answer:

> sin 36 * sin 72 * sin 108 * sin 144

=> sin 36 * sin 72 * sin (180 - 72) * sin (180 - 36)

=> sin 36 * sin 72 * sin 72 * sin 36

=> [sin 36 * sin 72]²

=>¼ [ 2sin 36 * sin 72]²

=>¼ [ cos 36 - cos 108]²

2sin(A)*sin(B) = [cos(A-B) - cos(A+B)]

=>¼ [ cos 36 + sin 18]² ; cos(108) = cos(90+18) = -sin(18)]

=>¼ [ (√5 + 1)/4 + (√5 - 1)/4]²

; put

cos 36 = (√5 + 1)/4 and sin 18 = (√5 - 1)/4

=>¼ [ √5/2 ]²

=> 5/16

Step-by-step explanation: