Prove Trigonometric Identities

[nycmath]

See attachment.

[Eigenvalue]

[tex]\frac{cos(x)}{1-sin(x)}[/tex]=[tex]\frac{1+sin(x)}{cos(x)}[/tex]

Cross multiply:
cos²(x)=(1+sin(x))(1-sin(x))

Use difference of squares to simplify:
cos²(x)=1-sin²(x)

Add sin²(x) to both sides:
sin²(x)+cos²(x)=1

sin²(x)+cos²(x)=1 is a Pythagorean identity
1=1