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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Factor

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

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

\frac{cos ( x )}{tan ( x )} = csc (x) - sin (x)

sec^{2} (θ) csc^{2} (θ) = sec^{2} (θ) + csc^{2} (θ)

sin (θ) tan (θ) + cos (θ) = \frac{1}{cos ( θ )}

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

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