1-ctg2a=(1+ctg2a)(sin2a-cos2a)

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вопрос

Let's simplify the given expression step by step:

First, expand the right side of the equation:
(1 + ctg2a)(sin2a - cos2a) = sin2a + sin2a · ctg2a - cos2a - cos2a · ctg2a
= sin2a + tan2a · sin2a - cos2a - cos2a · tan2a

Next, simplify the expression by using trigonometric identities:
tan2a = sin2a/cos2a
Therefore, tan2a · sin2a = sin2a/cos2a · sin2a = sin3a/cos2a

Applying this simplification:
= sin2a + sin3a/cos2a - cos2a - cos3a/cos2a
= sin2a + sin3a/cos2a - cos2a - sin2a
= sin2a - cos2a

Now, the left side of the equation:
1 - ctg2a

Since cotangent is the reciprocal of tangent, and tan2a = sin2a/cos2a, then cot2a = cos2a/sin2a. Therefore, ctg2a = cos2a/sin2a

Now, simplify the left side:
1 - ctg2a = 1 - cos2a/sin2a = (sin2a - cos2a)/sin2a

Since (sin2a - cos2a)/sin2a = sin2a - cos2a, the left side simplifies to:
1 - ctg2a = sin2a - cos2a

Therefore, the given equation simplifies to:
sin2a - cos2a = sin2a - cos2a

This confirms that the given expression is true.