Is cos(4θ)=cos^2(2θ)-sin^2(2θ) ?

The answer to whether cos(4θ)=cos^2(2θ)-sin^2(2θ) is True

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prove cos(4θ)=cos^2(2θ)-sin^2(2θ)

prove

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

True

Solution steps

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

Manipulating right side

cos^{2} (2 θ) - sin^{2} (2 θ)

Rewrite using trig identities

cos^{2} (2 θ) - sin^{2} (2 θ)

Use the Double Angle identity:

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

= cos (2 \cdot 2 θ)

Simplify

= cos (4 θ)

= cos (4 θ)

We showed that the two sides could take the same form

\Rightarrow True

Is cos(4θ)=cos^2(2θ)-sin^2(2θ) ?
The answer to whether cos(4θ)=cos^2(2θ)-sin^2(2θ) is True