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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Factor

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

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

Rewrite using trig identities

= - 2 cos^{2} (y) + 1

= 1 - 2 cos^{2} (y)

We showed that the two sides could take the same form

\Rightarrow True

1 - sin^{2} (- θ) = cos^{2} (θ)

tan (θ) csc^{2} (θ) - tan (θ) = cot (θ)

\frac{1 + sec ( θ )}{csc ( θ )} - tan (θ) = sin (θ)

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

cos (α + β) + cos (α - β) = 2 cos (α) cos (β)

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