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

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

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

prove

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

True

Solution steps

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

Manipulating right side

1 - sin (2 A)

Use the Pythagorean identity:

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

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

Use the Double Angle identity:

sin (2 A) = 2 sin (A) cos (A)

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

Apply Perfect Square Formula:

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

p ro v e - sec^{2} (x) = - \frac{1}{cos ^{2} ( x )}

p ro v e sin (2 x) = 3 sin (x)

p ro v e cos (\frac{2 π}{3}) = - \frac{1}{2}

p ro v e sin (x) \cdot cos (x) (tan (x) + cot (x)) = 1

p ro v e sin^{2} (x) = tan^{2} (x) cos^{2} (x)

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