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

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

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

prove

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

True

Solution steps

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

Manipulating right side

1 - sin (2 x)

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

csc (θ) tan (θ) = sec (θ)

\frac{1 + sin ( - x )}{1 + csc ( - x )} = - sin (x)

tan^{2} (x) + 1 = sec^{2} (x)

\frac{1 + csc ( x )}{sec ( x )} = cos (x) + cot (x)

cos (- x) = cos (x)

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