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

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

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

prove

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

True

Solution steps

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

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

We showed that the two sides could take the same form

\Rightarrow True

cos (- x) - sin (- x) = cos (x) + sin (x)

cos (θ) sec (θ) = 1

cos (a + b) + cos (a - b) = 2 cos (a) cos (b)

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

\frac{tan ( x ) cot ( x )}{cos ( x )} = sec (x)

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