(cos t)/(cos t/2 + sin t/2)

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вопрос

To simplify the expression (cos t)/(cos t/2 + sin t/2), we can use the double angle formulas for cosine and sine.

Recall that:
cos(2x) = 2cos^2(x) - 1
sin(2x) = 2sin(x)cos(x)

Let's use these two formulas to simplify the denominator (cos t/2 + sin t/2):

cos(t/2) = cos(2(t/4)) = 2cos^2(t/4) - 1
sin(t/2) = sin(2(t/4)) = 2sin(t/4)cos(t/4)

Now we substitute these expressions into the denominator:

cos(t/2) + sin(t/2) = 2cos^2(t/4) - 1 + 2sin(t/4)cos(t/4)
= 2(cos^2(t/4) + sin(t/4)cos(t/4)) - 1
= 2cos(t/4) - 1

Therefore, the expression becomes:

(cos t)/(2cos(t/4) - 1)

This is the simplified expression for (cos t)/(cos t/2 + sin t/2).