Is 4cos^3(θ)-3cos(θ)=cos(3θ) ?

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

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prove 4cos^3(θ)-3cos(θ)=cos(3θ)

prove

4 cos^{3} (θ) - 3 cos (θ) = cos (3 θ)

True

Solution steps

4 cos^{3} (θ) - 3 cos (θ) = cos (3 θ)

Manipulating right side

cos (3 θ)

Rewrite using trig identities

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

Use the Double Angle identity:

cos (2 θ) = 2 cos^{2} (θ) - 1

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

Use the Pythagorean identity:

cos^{2} (θ) + sin^{2} (θ) = 1

sin^{2} (θ) = 1 - cos^{2} (θ)

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

Expand

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

= 4 cos^{3} (θ) - 3 cos (θ)

We showed that the two sides could take the same form

\Rightarrow True

sec^{2} (x) + tan^{2} (x) sec^{2} (x) = sec^{4} (x)

cos (7 5^{\circ}) = cos (4 5^{\circ} + 3 0^{\circ})

\frac{1}{1 + cot ^{2} ( x )} = sin^{2} (x)

sec (θ) \cdot sin (θ) = tan (θ)

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

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