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

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

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

prove

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

True

Solution steps

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

Manipulating left 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

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

sec (A) - cos (A) = tan (A) sin (A)

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

cot (u) - csc (u) = \frac{sin ( u )}{1 + cos ( u )}

cos (θ) tan (- θ) = - sin (θ)

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