\frac{3}{1+tg^2\beta}+3sin^2\beta=
=\frac{3}{1+(\frac{sin\beta}{cos\beta})^2}+3sin\beta=
=\frac{3}{1+\frac{sin^2\beta}{cos^2\beta}}+3sin\beta=
=\frac{3}{\frac{cos^2\beta+sin^2\beta}{cos^2\beta}}+3sin\beta=
=\frac{3*cos^2\beta}{cos^2\beta+sin^2\beta}+3sin\beta=
=\frac{3cos^2\beta+3sin^2\beta}{cos^2\beta+sin^2\beta}=
=\frac{3(cos^2\beta+sin^2\beta)}{1}=3*1=3 co należało udowodnić