Simplify \sum_{k = 1}^n \tan(k) \tan(k - 1) by first

Gabriela Duarte

Gabriela Duarte

Answered question

2022-01-27

Simplify k=1ntan(k)tan(k1) by first proving tan(k)tan(k1)=tan(k)tan(k1)tan(1)1

Answer & Explanation

goleuedigdp

goleuedigdp

Beginner2022-01-28Added 7 answers

HINT:
Note that
tan1=tan(k(k1))=tanktan(k1)1+tanktan(k1)
from which the result follows.
The summation part is easy as the numerator is telescoping.
k=1n[tanktan(k1)tan11]=tanntan(n1)+tan(n1)tan(n2)++tan1tan0tan1n=tanntan1n
trasahed

trasahed

Beginner2022-01-29Added 14 answers

Let a=tanA b=tanB,,c=tan(AB)=ab1+ab so ab=abc1. Hence tanktan(k1)=tanktan(k1)tan11 and your sum is tanntan1n

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