Question

what is the value of 4cos^2(9)-3*4cos^2(27)-3

Answer 1

Answer:

(4cos²9°-3)(4cos²27°-3)=?

from identity we have cos3x=4cos³x-3cosx

which is the same as cosx(4cos²x-3)

so therefore to make the question in form of this identity we have to multiply by cosx/cosx

that is :

(4cos²9°-3)(4cos²27°-3)= cos9°(4cos²9°-3)/cos9° × cos27°(4cos²27°-3)/cos27

which is equal to

cos3(9°)/cos9° × cos3(27°)/cos27°

which is equal to (cos27°)/(cos9°) × (cos81°)/(cos27)

so cos 27° can easily cancel cos27°

so therefore we have cos81/cos9

from identity we have cosx=sin(90-x)

so cos81 can be expressed as sin(90-81) which is equal to sin9°

so (4cos²9°-3)(4cos²27°-3) = sin9°/cos9°

and we know that our (sinx/cosx)=tanx

so therefore sin9/cos9° = tan9°

so we can conclude that (4cos²9°-3)(4cos²27°-3) is equal to tan9°

Answer 2

Given:

A trigonometric expression 4cos^2(9)-3*4cos^2(27)-3.

To Find:

The value of the given expression.

Solution:

The given problem can be solved using the concepts

1. The given expression is 4cos^2(9)-3*4cos^2(27)-3.

2. According to the trigonometric formulae,

=> Cos3x = 4cos³x - 3cosx,

=> Cos3x = cosx( 4cos²x -3 ).

3. Conisder the given expression,

=> 4cos²(9)-3*4cos²(27)-3.

=> Multiply and divide the first term with cos9, multiply and divide the second term with cos27.

=> [(cos9)( 4cos²(9)-3 )]/cos9 x [(4cos²(27)-3)cos27]/cos27,

=> Simplify the above equation using the mentioned formula,

=> (4cos³(9)-3cos9) x  (4cos³(27) - 3cos27),

=> (cos27  x cos 81)/ ( cos9 x cos27 )

=> cos81/cos9,

=> sin9/cos9, ( Since sin(90-x) = cosx )

=> Tan 9.

Therefore, the value of the trigonometric expression 4cos²9 - 3 x 4cos²27 - 3 is Tan 9.