Is sin^4(θ)-cos^4(θ)=1-2cos^2(θ) ?

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

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prove sin^4(θ)-cos^4(θ)=1-2cos^2(θ)

prove

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

True

Solution steps

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

Manipulating left side

sin^{4} (θ) - cos^{4} (θ)

Factor

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

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

Rewrite using trig identities

= - 2 cos^{2} (θ) + 1

= 1 - 2 cos^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True

\frac{1 + tan ( x )}{1 - tan ( x )} + \frac{1 + cot ( x )}{1 - cot ( x )} = 0

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

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

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

(1 - sin^{2} (θ)) (1 + tan^{2} (θ)) = 1

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