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

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

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

prove

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

True

Solution steps

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

Manipulating right side

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

Apply Perfect Square Formula:

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

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

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

Use the Pythagorean identity:

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

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

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

Simplify

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

= cos^{4} (x)

We showed that the two sides could take the same form

\Rightarrow True

tan (x) \cdot cos (x) = sin (x)

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

\frac{sec ( x ) + tan ( x )}{csc ( x ) + 1} = tan (x)

tan (a) = \frac{sec ( a )}{csc ( a )}

sec (x) tan (x) cos (x) cot (x) = 1

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