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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

Simplify

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

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

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

Apply Perfect Square Formula:

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

a = sin (x), b = cos (x)

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

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

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

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

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

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

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

Simplify

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

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

Add similar elements:

cos^{2} (x) - cos^{2} (x) = 0

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

Add similar elements:

sin^{2} (x) - sin^{2} (x) = 0

= 2 sin (x) cos (x)

= 2 sin (x) cos (x)

= 2 sin (x) cos (x)

Use the Double Angle identity:

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

= sin (2 x)

= sin (2 x)

We showed that the two sides could take the same form

\Rightarrow True

cot (π)

cos (\frac{7 π}{2})

sin (2 5^{\circ})

arctan (\frac{1}{\frac{1}{3}})

sin (7 0^{\circ})

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