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

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

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

prove

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

True

Solution steps

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

Manipulating left side

(sin (x) - cos (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

\frac{2 cot ( x )}{csc ^{2} ( x )} = sin (2 x)

\frac{csc ^{2} ( x )}{1 + tan ^{2} ( x )} = cot^{2} (x)

sin (x) = \frac{2 tan ( \frac{x}{2} )}{1 + tan ^{2} ( \frac{x}{2} )}

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

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

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