Is cos(3b)=cos(b)(cos^2(b)-3sin^2(b)) ?

The answer to whether cos(3b)=cos(b)(cos^2(b)-3sin^2(b)) is True

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Popular Trigonometry >

prove cos(3b)=cos(b)(cos^2(b)-3sin^2(b))

Solution

prove

cos (3 b) = cos (b) (cos^{2} (b) - 3 sin^{2} (b))

Solution

True

Solution steps

cos (3 b) = cos (b) (cos^{2} (b) - 3 sin^{2} (b))

Manipulating left side

cos (3 b)

Rewrite using trig identities

cos (3 b)

Use the following identity:

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

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

Expand

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

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

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

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

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)

= cos^{2 + 1} (x)

Add the numbers:

2 + 1 = 3

= cos^{3} (x)

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

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

Add similar elements:

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

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

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

= cos^{3} (b) - 3 sin^{2} (b) cos (b)

= cos^{3} (b) - 3 sin^{2} (b) cos (b)

Manipulating right side

cos (b) (cos^{2} (b) - 3 sin^{2} (b))

Expand

(cos^{2} (b) - 3 sin^{2} (b)) cos (b) : cos^{3} (b) - 3 sin^{2} (b) cos (b)

(cos^{2} (b) - 3 sin^{2} (b)) cos (b)

= cos (b) (cos^{2} (b) - 3 sin^{2} (b))

Apply the distributive law:

a (b - c) = ab - a c

a = cos (b), b = cos^{2} (b), c = 3 sin^{2} (b)

= cos (b) cos^{2} (b) - cos (b) \cdot 3 sin^{2} (b)

= cos^{2} (b) cos (b) - 3 sin^{2} (b) cos (b)

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

cos^{2} (b) cos (b)

Apply exponent rule:

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

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

= cos^{2 + 1} (b)

Add the numbers:

2 + 1 = 3

= cos^{3} (b)

= cos^{3} (b) - 3 sin^{2} (b) cos (b)

= cos^{3} (b) - 3 cos (b) sin^{2} (b)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is cos(3b)=cos(b)(cos^2(b)-3sin^2(b)) ?
The answer to whether cos(3b)=cos(b)(cos^2(b)-3sin^2(b)) is True