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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

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

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

tan (θ) + cot (θ) = csc (θ) sec (θ)

2 sin (x) + 1 = 3 csc (x)

sec (x) cot (x) = \frac{1}{sin ( x )}

sec (2 θ) = \frac{sec ^{2} ( θ ) csc ^{2} ( θ )}{csc ^{2} ( θ ) - sec ^{2} ( θ )}

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