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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

= cos (2 \cdot 2 x)

Simplify

= cos (4 x)

= cos (4 x)

We showed that the two sides could take the same form

\Rightarrow True

p ro v e \frac{sec ^{2} ( x ) - 1}{sin ( x )} = \frac{sin ( x )}{1 - sin ^{2} ( x )}

p ro v e sin (\frac{π}{4} + x) + sin (\frac{π}{4} - x) = 2 cos (x)

p ro v e sin (θ) sec (θ) cot (θ) = 1

p ro v e 1 + sec^{2} (θ) sin^{2} (θ) = sec^{2} (θ)

p ro v e 3 sin^{2} (x) + 4 cos^{2} (x) = 3 + cos^{2} (x)

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