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

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

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prove 1-2cos^2(x)+cos^4(x)=sin^4(x)

prove

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

True

Solution steps

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

Manipulating left side

1 - 2 cos^{2} (x) + cos^{4} (x)

Apply Perfect Square Formula:

a^{2} - 2 ab + b^{2} = (a - b)^{2}

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

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

Use the Pythagorean identity:

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

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

= (sin^{2} (x))^{2}

Simplify

(sin^{2} (x))^{2} : sin^{4} (x)

= sin^{4} (x)

We showed that the two sides could take the same form

\Rightarrow True

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

tan (x) + cot (x) = (sec (x)) (csc (x))

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

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

sec (y) cos (y) = 1

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