How do you prove? (cos^2x+tan^2x-1)/sin^2x=tan^2x

1 Answer
Jan 29, 2018

We seek to prove that:

# (cos^2x+tan^2x-1)/sin^2x=tan^2x #

Consider the LHS:

# LHS = (cos^2x+tan^2x-1)/sin^2x #

# \ \ \ \ \ \ \ \ = (1-sin^2x+sin^2x/cos^2x-1)/sin^2x #

# \ \ \ \ \ \ \ \ = (sin^2x/cos^2x-sin^2x)/sin^2x #

# \ \ \ \ \ \ \ \ = 1/cos^2x-1 #

# \ \ \ \ \ \ \ \ = (1-cos^2x)/cos^2x #

# \ \ \ \ \ \ \ \ = sin^2x/cos^2x #

# \ \ \ \ \ \ \ \ = tan^2x \ \ \ # QED