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^{2} (x))^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

Multiply the numbers:

2 \cdot 2 = 4

= sin^{4} (x)

= sin^{4} (x)

We showed that the two sides could take the same form

\Rightarrow True

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