Question

Uzasadnij tożsamość:

a)

b)

c)

Answer 1

A)
tg²α - sin²α = tg²α × sin²α
L - lewa strona równania
P - prawa strona równania
L = tg²α - sin²α =
= sin²α / cos²α - sin²α =
= (sin²α - sin²αcos²α) / cos²α =
= [sin²α(1 - cos²α)] / cos²α =
= sin²αsin²α / cos²α =
= tg²α×cos²α =
= P
b)
L = cosα × ctgα + sinα =
= cosα × cosα/sinα + sinα =
= cos²α / sinα + sinα =
= (cos²α + sin²α) / sinα =
= 1 / sinα =
= P
c)
L = cosα / (sinα -1) + tgα=
= cosα / (sinα - 1) + sinα / cosα =
= {cos²α + (sinα - 1) × sinα] / [(sinα - 1) × cosα] =
= (cos²α + sin²α - sinα) / [(sinα - 1) ×cosα) =
= (1 - sinα) / [(sinα - 1) × cosα] =
= [-(sinα - 1)] / [(sinα - 1) × cosα] =
= -1 / cosα =
= P