Question

Sprawdź tożsamości:

a) sin x / (1+ cos x) + (1 + cos x) / sinx = 2/sinx

b) [(1- sinx)/ sinx] * [(1+ sinx)/cosx] = ctgx

Answer 1

a)

sin x / (1+ cos x) + (1 + cos x) / sinx = 2/sinx

L = sin x / (1+ cos x) + (1 + cos x) / sinx =

= sin²x + (1+ cos x) ² /sinx(1 + cos x) =

= sin²x + 1 +2cosx + cos²x / sinx(1 + cos x) =

= 2 + 2cosx /sinx(1 + cos x) =

= 2(1 + cosx) / sinx(1 + cos x) =

= 2/sinx = P


b)

[(1- sinx)/ sinx] * [(1+ sinx)/cosx] = ctgx

L = [(1- sinx)/ sinx] * [(1+ sinx)/cosx] =

= 1 - sin²x / sinx*cosx =

= cos²x / sinx*cosx =

= cosx / sinx =

= ctgx = P