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

Apply Perfect Square Formula:

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

a = cos (x), b = sin (x)

= 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

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

Use the Double Angle identity:

2 sin (x) cos (x) = sin (2 x)

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

Use the Pythagorean identity:

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

= 1 - sin (2 x)

= 1 - sin (2 x)

We showed that the two sides could take the same form

\Rightarrow True

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