Is 2sin(2θ)(1-2sin^2(θ))=sin(4θ) ?

The answer to whether 2sin(2θ)(1-2sin^2(θ))=sin(4θ) is True

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prove 2sin(2θ)(1-2sin^2(θ))=sin(4θ)

prove

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

True

Solution steps

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

Manipulating left side

2 sin (2 θ) (1 - 2 sin^{2} (θ))

Rewrite using trig identities

2 sin (2 θ) (1 - 2 sin^{2} (θ))

Use the Double Angle identity:

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

= 2 sin (2 θ) cos (2 θ)

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

= sin (2 \cdot 2 θ)

Simplify

= sin (4 θ)

= sin (4 θ)

We showed that the two sides could take the same form

\Rightarrow True

Is 2sin(2θ)(1-2sin^2(θ))=sin(4θ) ?
The answer to whether 2sin(2θ)(1-2sin^2(θ))=sin(4θ) is True