Question

1-(cosx-sinx)²=sin2x

Answer 1

Решение во вложении. Удачи.
Один раз подчеркнул : cosx²+sinx²=1
Два раза подчеркнул : 2sinx*cosx=sin2x

Answer 2

Let's verify the given trigonometric identity step by step:

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

Step 1: Expand the square term on the left side.

Recall that (a - b)^2 = a^2 - 2ab + b^2.

So, (cos(x) - sin(x))^2 = cos^2(x) - 2*cos(x)*sin(x) + sin^2(x).

Step 2: Substitute the expanded square term back into the equation.

1 - (cos^2(x) - 2*cos(x)*sin(x) + sin^2(x)) = sin(2x).

Step 3: Simplify the left-hand side.

Recall that 1 - cos^2(x) = sin^2(x) (using the Pythagorean identity).

So, the equation becomes: sin^2(x) + 2*cos(x)*sin(x) - sin^2(x) = sin(2x).

Step 4: Combine like terms on the left side.

2*cos(x)*sin(x) = sin(2x).

Step 5: Rewrite sin(2x) using a double angle identity.

The double angle identity for sin is: sin(2x) = 2*sin(x)*cos(x).

So, the equation becomes: 2*cos(x)sin(x) = 2sin(x)*cos(x).

Step 6: Divide both sides by 2*sin(x)*cos(x) (common factor).

This simplifies to: 1 = 1.

The equation is true.

Therefore, 1 - (cos(x) - sin(x))^2 = sin(2x) is indeed an identity; it holds true for all values of x.