Is cos^3(A)= 1/4 (cos(3A)+3cos(A)) ?

The answer to whether cos^3(A)= 1/4 (cos(3A)+3cos(A)) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove cos^3(A)= 1/4 (cos(3A)+3cos(A))

Solution

prove

cos^{3} (A) = \frac{1}{4} (cos (3 A) + 3 cos (A))

Solution

True

Solution steps

cos^{3} (A) = \frac{1}{4} (cos (3 A) + 3 cos (A))

Manipulating right side

\frac{1}{4} (cos (3 A) + 3 cos (A))

Rewrite using trig identities

\frac{1}{4} (cos (3 A) + 3 cos (A))

Use the following identity:

cos (3 x) = 4 cos^{3} (x) - 3 cos (x)

cos (3 x)

Rewrite using trig identities

cos (3 x)

Rewrite as

= cos (2 x + x)

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (2 x) cos (x) - sin (2 x) sin (x)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= cos (2 x) cos (x) - 2 sin (x) cos (x) sin (x)

Simplify

cos (2 x) cos (x) - 2 sin (x) cos (x) sin (x) : cos (x) cos (2 x) - 2 sin^{2} (x) cos (x)

cos (2 x) cos (x) - 2 sin (x) cos (x) sin (x)

2 sin (x) cos (x) sin (x) = 2 sin^{2} (x) cos (x)

2 sin (x) cos (x) sin (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

= 2 cos (x) sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= 2 cos (x) sin^{2} (x)

= cos (x) cos (2 x) - 2 sin^{2} (x) cos (x)

= cos (x) cos (2 x) - 2 sin^{2} (x) cos (x)

= cos (x) cos (2 x) - 2 sin^{2} (x) cos (x)

Use the Double Angle identity:

cos (2 x) = 2 cos^{2} (x) - 1

= (2 cos^{2} (x) - 1) cos (x) - 2 sin^{2} (x) cos (x)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= (2 cos^{2} (x) - 1) cos (x) - 2 (1 - cos^{2} (x)) cos (x)

Expand

(2 cos^{2} (x) - 1) cos (x) - 2 (1 - cos^{2} (x)) cos (x) : 4 cos^{3} (x) - 3 cos (x)

(2 cos^{2} (x) - 1) cos (x) - 2 (1 - cos^{2} (x)) cos (x)

= cos (x) (2 cos^{2} (x) - 1) - 2 cos (x) (1 - cos^{2} (x))

Expand

cos (x) (2 cos^{2} (x) - 1) : 2 cos^{3} (x) - cos (x)

cos (x) (2 cos^{2} (x) - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = cos (x), b = 2 cos^{2} (x), c = 1

= cos (x) 2 cos^{2} (x) - cos (x) 1

= 2 cos^{2} (x) cos (x) - 1 cos (x)

Simplify

2 cos^{2} (x) cos (x) - 1 \cdot cos (x) : 2 cos^{3} (x) - cos (x)

2 cos^{2} (x) cos (x) - 1 cos (x)

2 cos^{2} (x) cos (x) = 2 cos^{3} (x)

2 cos^{2} (x) cos (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cos^{2} (x) cos (x) = cos^{2 + 1} (x)

= 2 cos^{2 + 1} (x)

Add the numbers:

2 + 1 = 3

= 2 cos^{3} (x)

1 \cdot cos (x) = cos (x)

1 cos (x)

Multiply:

1 \cdot cos (x) = cos (x)

= cos (x)

= 2 cos^{3} (x) - cos (x)

= 2 cos^{3} (x) - cos (x)

= 2 cos^{3} (x) - cos (x) - 2 (1 - cos^{2} (x)) cos (x)

Expand

- 2 cos (x) (1 - cos^{2} (x)) : - 2 cos (x) + 2 cos^{3} (x)

- 2 cos (x) (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 2 cos (x), b = 1, c = cos^{2} (x)

= - 2 cos (x) 1 - (- 2 cos (x)) cos^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 2 \cdot 1 cos (x) + 2 cos^{2} (x) cos (x)

Simplify

- 2 \cdot 1 \cdot cos (x) + 2 cos^{2} (x) cos (x) : - 2 cos (x) + 2 cos^{3} (x)

- 2 \cdot 1 cos (x) + 2 cos^{2} (x) cos (x)

2 \cdot 1 \cdot cos (x) = 2 cos (x)

2 \cdot 1 cos (x)

Multiply the numbers:

2 \cdot 1 = 2

= 2 cos (x)

2 cos^{2} (x) cos (x) = 2 cos^{3} (x)

2 cos^{2} (x) cos (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cos^{2} (x) cos (x) = cos^{2 + 1} (x)

= 2 cos^{2 + 1} (x)

Add the numbers:

2 + 1 = 3

= 2 cos^{3} (x)

= - 2 cos (x) + 2 cos^{3} (x)

= - 2 cos (x) + 2 cos^{3} (x)

= 2 cos^{3} (x) - cos (x) - 2 cos (x) + 2 cos^{3} (x)

Simplify

2 cos^{3} (x) - cos (x) - 2 cos (x) + 2 cos^{3} (x) : 4 cos^{3} (x) - 3 cos (x)

2 cos^{3} (x) - cos (x) - 2 cos (x) + 2 cos^{3} (x)

Group like terms

= 2 cos^{3} (x) + 2 cos^{3} (x) - cos (x) - 2 cos (x)

Add similar elements:

2 cos^{3} (x) + 2 cos^{3} (x) = 4 cos^{3} (x)

= 4 cos^{3} (x) - cos (x) - 2 cos (x)

Add similar elements:

- cos (x) - 2 cos (x) = - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x)

= \frac{1}{4} (4 cos^{3} (A) - 3 cos (A) + 3 cos (A))

Simplify

\frac{1}{4} (4 cos^{3} (A) - 3 cos (A) + 3 cos (A)) : cos^{3} (A)

\frac{1}{4} (4 cos^{3} (A) - 3 cos (A) + 3 cos (A))

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot ( 4 cos ^{3} ( A ) - 3 cos ( A ) + 3 cos ( A ) )}{4}

1 \cdot (4 cos^{3} (A) - 3 cos (A) + 3 cos (A)) = 4 cos^{3} (A)

1 \cdot (4 cos^{3} (A) - 3 cos (A) + 3 cos (A))

Add similar elements:

- 3 cos (A) + 3 cos (A) = 0

= 1 \cdot 4 cos^{3} (A)

Multiply the numbers:

1 \cdot 4 = 4

= 4 cos^{3} (A)

= \frac{4 cos ^{3} ( A )}{4}

Divide the numbers:

\frac{4}{4} = 1

= cos^{3} (A)

= cos^{3} (A)

= cos^{3} (A)

We showed that the two sides could take the same form

\Rightarrow True

Popular Examples

prove (sin(6x)-sin(2x))/(cos(6x)-cos(2x))=-(cos(4x))/(sin(4x))prove

\frac{sin ( 6 x ) - sin ( 2 x )}{cos ( 6 x ) - cos ( 2 x )} = - \frac{cos ( 4 x )}{sin ( 4 x )}

prove sin(x)=-1/2 prove

sin (x) = - \frac{1}{2}

prove 1/(sin(x)+1)+1/(csc(+1))=1 prove

\frac{1}{sin ( x ) + 1} + \frac{1}{csc ( + 1 )} = 1

prove cos(x)=sin(x-pi)prove

cos (x) = sin (x - π)

prove cos(φ+a)-cos(φ-a)=-2sin(φ)sin(a)prove

cos (φ + a) - cos (φ - a) = - 2 sin (φ) sin (a)

Frequently Asked Questions (FAQ)

Is cos^3(A)= 1/4 (cos(3A)+3cos(A)) ?
The answer to whether cos^3(A)= 1/4 (cos(3A)+3cos(A)) is True