sin,cos,tan,sec,csc,cot數學難題

[匿名]

2012-10-03 5:12 pm

Prove the given identities. help me out plz.

1. (cotx / 1-tanx) + ( tanx / 1-cotx) = 1 + secxcscx

2. (cosx - siny / cosy - sinx) = (cosy + sinx / cosx + siny)

回答 (1)

暴雨

2012-10-03 7:40 pm

✔ 最佳答案

1. LHS
= cot x / (1 - tan x) + tan x / (1 - cot x)
= cos^2 x / [sin x (cos x - sin x)] + sin^2 / [cos x (sin x - cos x)]
= (cos^3 x - sin^3 x) / [sin x cos x (cos x - sin x)]
= (cos^2 x + sinx x cos x + sin^2) / (sin x cos x)
= 1 + 1 / (sin x cos x)
= 1 + sec x csc x
= RHS

2. LHS
(cos x - sin y) / (cos y - sin x)
= [(cos x - sin y)(cos x + sin y)] / [(cos y - sin x)(cos x + sin y)]
= (cos^2 x - sin^2 y) / [(cos y - sin x)(cos x + sin y)]
= (1 - sin^2 x - 1 + cos^2 y) / [(cos y - sin x)(cos x + sin y)]
= (cos^2 y - sin^2 x) / [(cos y - sin x)(cos x + sin y)]
= (cos y + sin x) / (cos x + sin y)
= RHS

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