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

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

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

prove

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

True

Solution steps

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

Manipulating left side

cos (3 x)

Rewrite using trig identities

= 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)

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

We showed that the two sides could take the same form

\Rightarrow True

cos (θ) = sin (\frac{π}{2} + θ)

1 - 8 sin^{2} (θ) cos^{2} (θ) = cos (4 θ)

\frac{cos ( b )}{sin ( b )} \cdot sin (b) = cos (b)

- 50 cos (4 t) + 50 = 100 sin^{2} (2 t)

\frac{7 sec ( θ ) - 7}{1 - cos ( θ )} = 7 sec (θ)

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