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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Expand

(cos (x) - sin (x))^{2} : cos^{2} (x) - 2 cos (x) sin (x) + sin^{2} (x)

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

Rewrite using trig identities

= 1 - sin (2 x)

We showed that the two sides could take the same form

\Rightarrow True

cot (x) tan (x) = 1

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

tan (u) cot (u) - sin^{2} (u) = cos^{2} (u)

1 - 2 sin^{2} (2 x) = 8 cos^{4} (x) - 8 cos^{2} (x) + 1

sin (\frac{π}{4}) = \frac{2}{2}

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