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

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

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

prove

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

True

Solution steps

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

Manipulating right side

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

Factor

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

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

Rewrite using trig identities

= cos (2 x)

We showed that the two sides could take the same form

\Rightarrow True

csc (\frac{π}{2} - u) = sec (u)

- 2 sin (k - l) sin (k + l) = cos (2 k) - cos (2 l)

tan^{2} (x) + sec^{2} (x) = cot^{2} (x)

csc (x) - 1 = \frac{1 - sin ( x )}{sin ( x )}

tan (x) sin (x) = \frac{sec ^{2} ( x ) - 1}{sec ( x )}

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