(cos7pi/10 + sin7pi/10) / (cospi/5 + sinpi/5)

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вопрос

To simplify this expression, we first need to use the double angle formula to rewrite the trigonometric functions.

Let's start by rewriting cos(7π/10) and sin(7π/10) in terms of cos(π/5) and sin(π/5):

cos(7π/10) = cos(2π/2 + 3π/10) = cos(3π/10) = cos(π/2 - π/10) = sin(π/10)
sin(7π/10) = sin(2π/2 + 3π/10) = sin(3π/10) = sin(π/2 - π/10) = cos(π/10)

Now, we rewrite the expression in terms of sin(π/10) and cos(π/5):

(cos(7π/10) + sin(7π/10)) / (cos(π/5) + sin(π/5))
= (sin(π/10) + cos(π/10)) / (cos(π/5) + sin(π/5))

Now, we can use the trigonometric identity sin(A) + cos(A) = √2sin(A + π/4) to rewrite the expression:

= √2sin(π/10 + π/4) / √2sin(π/5 + π/4)
= sin(3π/20) / sin(9π/20)
= sin(3π/20) / sin(π - 3π/20)
= sin(3π/20) / sin(17π/20)

Therefore, the simplified form of the expression (cos(7π/10) + sin(7π/10)) / (cos(π/5) + sin(π/5)) is sin(3π/20) / sin(17π/20).