Is cos(3a)=4cos^3(a)-3cos(a) ?

The answer to whether cos(3a)=4cos^3(a)-3cos(a) is True

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prove cos(3a)=4cos^3(a)-3cos(a)

Solution

prove

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

Solution

True

Solution steps

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

Manipulating left side

cos (3 a)

Rewrite using trig identities

cos (3 a)

Rewrite as

= cos (2 a + a)

Use the Angle Sum identity:

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

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

Use the Double Angle identity:

sin (2 a) = 2 sin (a) cos (a)

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

Simplify

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

cos (2 a) cos (a) - 2 sin (a) cos (a) sin (a)

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

2 sin (a) cos (a) sin (a)

Apply exponent rule:

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

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

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

Add the numbers:

1 + 1 = 2

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

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

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

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

Use the Double Angle identity:

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

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

Use the Pythagorean identity:

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

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

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

Expand

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

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

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

Simplify

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

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

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

2 cos^{2} (a) cos (a)

Apply exponent rule:

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

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

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

Add the numbers:

2 + 1 = 3

= 2 cos^{3} (a)

1 \cdot cos (a) = cos (a)

1 cos (a)

Multiply:

1 \cdot cos (a) = cos (a)

= cos (a)

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

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Apply minus-plus rules

- (- a) = a

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

Simplify

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

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

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

2 \cdot 1 cos (a)

Multiply the numbers:

2 \cdot 1 = 2

= 2 cos (a)

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

2 cos^{2} (a) cos (a)

Apply exponent rule:

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

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

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

Add the numbers:

2 + 1 = 3

= 2 cos^{3} (a)

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

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

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

Add similar elements:

- cos (a) - 2 cos (a) = - 3 cos (a)

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is cos(3a)=4cos^3(a)-3cos(a) ?
The answer to whether cos(3a)=4cos^3(a)-3cos(a) is True