Is 4-8sin^2(x)=4cos(2x) ?

The answer to whether 4-8sin^2(x)=4cos(2x) is True

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prove 4-8sin^2(x)=4cos(2x)

prove

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

True

Solution steps

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

Manipulating left side

4 - 8 sin^{2} (x)

Factor

4 - 8 sin^{2} (x) : 4 (1 - 2 sin^{2} (x))

4 - 8 sin^{2} (x)

Rewrite as

= 4 \cdot 1 - 4 \cdot 2 sin^{2} (x)

Factor out common term

4

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

= (1 - 2 sin^{2} (x)) \cdot 4

Manipulating right side

4 cos (2 x)

Rewrite using trig identities

4 cos (2 x)

Use the Double Angle identity:

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

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

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

We showed that the two sides could take the same form

\Rightarrow True