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

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

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

prove

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

True

Solution steps

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

Manipulating right side

- sin (2 x) + 1

Use the Pythagorean identity:

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

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

Use the Double Angle identity:

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

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

Apply Perfect Square Formula:

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

sec^{2} (A) - cos (A) sec (A) = tan^{2} (A)

\frac{csc ( - x )}{sec ( - x )} = - tan (\frac{π}{2} - x)

sin (θ) + \frac{cos ^{2} ( θ )}{sin ( θ )} = csc (θ)

1 - cot (x) = (1 - 2 cot^{2} (x))

\frac{cos ( - x ) tan ( - x )}{sin ( x )} = - 1

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