(sin 5п/18+sin 11п/9)/(cos 5п/18+cos 11п/9)

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вопрос

To simplify this expression, we can use the trigonometric identity:

cos(a) + cos(b) = 2cos((a+b)/2)cos((a-b)/2)

sin(a) + sin(b) = 2cos((a+b)/2)sin((a-b)/2)

Given that a = 5π/18 and b = 11π/9, we can substitute these values into the identities:

cos(5π/18) + cos(11π/9) = 2cos((5π/18 + 11π/9)/2)cos((11π/9 - 5π/18)/2)
= 2cos(25π/18)cos(7π/18)

sin(5π/18) + sin(11π/9) = 2cos((5π/18 + 11π/9)/2)sin((11π/9 - 5π/18)/2)
= 2cos(25π/18)sin(7π/18)

Now, the expression becomes:

(2cos(25π/18)sin(7π/18)) / (2cos(25π/18)cos(7π/18))

The cosine terms cancel out:

sin(7π/18) / cos(7π/18)

= tan(7π/18)

Therefore, the simplified expression is tan(7π/18).