Is cos^4(θ)-sin^4(θ)=2cos^2(θ)-1 ?

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

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prove cos^4(θ)-sin^4(θ)=2cos^2(θ)-1

prove

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

True

Solution steps

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

Manipulating left side

cos^{4} (θ) - sin^{4} (θ)

Factor

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

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

Rewrite using trig identities

= 2 cos^{2} (θ) - 1

We showed that the two sides could take the same form

\Rightarrow True

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

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

cos (4 θ) = 8 cos^{4} (θ) - 8 cos^{2} (θ) + 1

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

sin (2 A) = 2 sin (A) cos (A)

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