Simplify ∑k=1ntan⁡(k)tan⁡(k−1) by first proving tan⁡(k)tan⁡(k−1)=tan⁡(k)−tan⁡(k−1)tan⁡(1)−1

Gabriela Duarte

Gabriela Duarte

Answered

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

Expert

2022-01-28Added 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

Expert

2022-01-29Added 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

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?