Question

cos 6 sin 24 cos 72 =••••••••?​

Answer 1

Answer:

cos 6 sin 24 cos 72

=(1/2)*2 cos 6 sin 24 cos 72

=(1/2) [sin (6+24) - sin (6-24)] * cos 72

=(1/2) [sin 30 - sin (-18)] * cos 72

=(1/2) [( 1/2) + sin 18] * cos 72................................equ (i)

We know,

Sin 72 = 2 sin 36* cos 36

Cos ( 90-72) = 2 ( 2 sin 18 * cos 18) * ( 1 - 2 sin²18)

Cos 18 = 4 sin18.cos18 * (1 - 2 sin²18)

1 = 4sin18 * (1 - 2 sin²18)........................................equ (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 = ( √5 - 1 ) / 4    [ Hint : compare given equations with ax² + bx + c = 0 )

sin18  =  ( √5 - 1 ) / 4

Put values of sin 18 in equ(i)

= (1/2) [ 1/2 + ( √5 - 1 ) / 4] * cos 72

= (1/2) [ 1/2 + ( √5 - 1 ) / 4] * sin (90 - 72 )

= (1/2) [ 1/2 + ( √5 - 1 ) / 4] * sin 18

= (1/2) [ 1/2 + ( √5 - 1 ) / 4] * ( √5 - 1 ) / 4

= (1/2) [( √5 + 1 )/4] * ( √5 - 1 ) / 4

= (5 - 1 )/ 32

= 4/32

=1/8  Ans.

Step-by-step explanation: