(cosx-sin)^2=1-2sin2x

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вопрос

To simplify the left hand side of the equation, we need to expand the square of the expression (cosx - sin).

(cosx - sin)^2 = (cosx - sin)(cosx - sin)
Expanding this using FOIL (First, Outer, Inner, Last) method, we get:
= cosx cosx - cosx sin - sin cosx + sin sin
= cos^2(x) - 2sin(x)cos(x) + sin^2(x)

Now, using trigonometric identities:
cos^2(x) + sin^2(x) = 1
2sin(x)cos(x) = sin(2x)

Therefore, cos^2(x) - 2sin(x)cos(x) + sin^2(x) = 1 - sin(2x)
Hence, the left hand side of the equation simplifies to:
(cosx - sin)^2 = 1 - sin(2x)

And the equation becomes:
1 - sin(2x) = 1 - 2sin(2x)
This is the simplified version of the given equation.