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

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

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

prove

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

True

Solution steps

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

Manipulating left side

2 cos (2 x)

Rewrite using trig identities

2 cos (2 x)

Use the Double Angle identity:

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

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

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

Manipulating right side

4 cos^{2} (x) - 2

Factor

- 2 + 4 cos^{2} (x) : 2 (- 1 + 2 cos^{2} (x))

- 2 + 4 cos^{2} (x)

Rewrite as

= - 2 \cdot 1 + 2 \cdot 2 cos^{2} (x)

Factor out common term

2

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

= (- 1 + 2 cos^{2} (x)) \cdot 2

We showed that the two sides could take the same form

\Rightarrow True

tan (\frac{π}{4} - θ) = \frac{cos ( 2 θ )}{1 + sin ( 2 θ )}

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

sec (2 π) = 1 - tan^{2} (2 π)

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

\frac{sin ( x )}{sin ( 2 x )} = \frac{sec ( x )}{2}