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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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)

We showed that the two sides could take the same form

\Rightarrow True

p ro v e (1 + cot^{2} (x)) tan (x) = csc (x) sec (x)

p ro v e sin (a + b) + sin (a - b) = 2 sin (a) cos (b)

p ro v e (cos (x)) (1 + tan (x))^{2} = sec (x) + 2 sin (x)

p ro v e \frac{1 + tan ^{2} ( x )}{2 tan ( x )} = csc (2 x)

p ro v e cos (3 θ) = cos^{3} (θ) - 3 cos (θ) sin^{2} (θ)

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