Is sin^4(u)-cos^4(u)=2sin^2(u)-1 ?

The answer to whether sin^4(u)-cos^4(u)=2sin^2(u)-1 is True

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prove sin^4(u)-cos^4(u)=2sin^2(u)-1

prove

sin^{4} (u) - cos^{4} (u) = 2 sin^{2} (u) - 1

True

Solution steps

sin^{4} (u) - cos^{4} (u) = 2 sin^{2} (u) - 1

Manipulating left side

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

Factor

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

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

Rewrite using trig identities

= - 1 + 2 sin^{2} (u)

= 2 sin^{2} (u) - 1

We showed that the two sides could take the same form

\Rightarrow True

tan^{4} (x) - sec^{4} (x) = - 2 sec^{2} (x) + 1

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

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

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

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

Is sin^4(u)-cos^4(u)=2sin^2(u)-1 ?
The answer to whether sin^4(u)-cos^4(u)=2sin^2(u)-1 is True