Question

if x=30 then prove 4cos^2(x)-3cos(x)=cos3(x)

Answer 1

Explanation:

So we will be proving that cos3x=4cos3x−3cosx

[1]XXcos3x

[2]XX=cos(x+2x)

Angle sum identity: cos(α+β)=cosαcosβ−sinαsinβ

[3]XX=cosxcos2x−sinxsin2x

Double angle identity: cos2α=2cos2α−1

[4]XX=cosx(2cos2x−1)−sinxsin2x

[5]XX=2cos3x−cosx−sinxsin2x

Double angle identity: sin2α=2sinαcosα

[6]XX=2cos3x−cosx−sinx(2sinxcosx)

[7]XX=2cos3x−cosx−sin2x(2cosx)

Pythagorean identity: sin2α=1−cos2α

[8]XX=2cos3x−cosx−(1−cos2x)(2cosx)

[9]XX=2cos3x−cosx−(2cosx−2cos3x)

[10]XX=2cos3x−cosx−2cosx+2cos3x

Combine like terms.

[11]XX=4cos3x−3cosx

∴cos3x=4cos3x−3cosx

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Answer 2

Answer:

0 = 0, LHS = RHS

Step-by-step explanation:

substituting x = 30° in LHS

we will get,

LHS = cos 3 (30°) = cos 90° = 0

now, substituting x = 30° in RHS

RHS = 4 cos³*30° - 3 cos 30°

RHS = 4 (√3/2)³ - 3 ( √3 / 2 )

RHS = 4 ( 3√3 / 8 ) - (3√3 / 2)

RHS = ( 3√3/2) - (3√3/2)

RHS = 0