Q. Evaluate: Cos6° × sin24° × cos72°

Soln:

   Cos6° × sin24° × cos72°

= (1/2) * 2 Cos6° × sin24° × cos72°

= (1/2) [sin(6+24) – sin(6-24)] * cos72

= (1/2) [sin30 – sin(-18)] * cos72

= (1/2) [1/2 + sin18] * cos72 …….eqn(i)

We know,

Sin72 = 2 sin36 * cos36

cos(90-72) = 2 (2 sin18 * cos18) * (1 – 2sin²18)

cos18 = 4sin18.cos18 * (1 – 2sin²18)

1 = 4sin18 * (1 – 2sin²18) …………eqn(ii)

Let sin18 = x, then

1 = 4x * (1 – 2x²)

8x³ – 4x + 1 = 0 [Hint: use synthetic division formula to factorize.]

(2x – 1)(4x² + 2x – 1) = 0

Here,

x = 1/2

sin18 = 1/2  is not possible .

4x² + 2x – 1 = 0 gives

x =  (Ö5 – 1)/4 (Hint: compare given equation with ax² + bx + c = 0 )

sin18 =  (Ö5 – 1)/4

put value of sin18 in eqn(i)

= (1/2) [1/2 +  (Ö5 – 1)/4] * cos72

= (1/2) [1/2 +  (Ö5 – 1)/4] * sin(90 – 72)

= (1/2) [1/2 +  (Ö5 – 1)/4] * sin18

= (1/2) [1/2 +  (Ö5 – 1)/4] *  (Ö5 – 1)/4

= (1/2) [(Ö5 + 1)/4] * (Ö5 – 1)/4

= (5 – 1)/32

= 4/32

= 1/ 8 Ans.