(sin(pi/4-a)+cos(pi/4-a))/(sin(pi/4-a)-cos(pi/4-a)) упростить

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вопрос

To simplify the given expression, we can first simplify the numerator and denominator separately.

Numerator:
sin(pi/4 - a) + cos(pi/4 - a)
Using the angle subtraction formula for sine and cosine:
sin(pi/4)cos(a) - cos(pi/4)sin(a) + cos(pi/4)cos(a) + sin(pi/4)sin(a)
= sqrt(2)/2 cos(a) - sqrt(2)/2 sin(a) + sqrt(2)/2 cos(a) + sqrt(2)/2 sin(a)
= sqrt(2) * cos(a)

Denominator:
sin(pi/4 - a) - cos(pi/4 - a)
Using the angle subtraction formula for sine and cosine:
sin(pi/4)cos(a) - cos(pi/4)sin(a) - cos(pi/4)cos(a) - sin(pi/4)sin(a)
= sqrt(2)/2 cos(a) - sqrt(2)/2 sin(a) - sqrt(2)/2 cos(a) - sqrt(2)/2 sin(a)
= -sqrt(2) * sin(a)

Therefore, the simplified expression is:
(sqrt(2) cos(a)) / (-sqrt(2) sin(a))
= -cot(a)