Is #tan^2x -= sec^2x - 1# an identity?

2 Answers
Jan 9, 2018

True

Explanation:

Start with the well known pythagorean identity:

# sin^2x + cos^2x -= 1 #

This is readily derived directly from the definition of the basic trigonometric functions #sin# and #cos# and Pythagoras's Theorem.

Divide both side by #cos^2x# and we get:

# sin^2x/cos^2x + cos^2x/cos^2x -= 1/cos^2x #

# :. tan^2x + 1 -= sec^2x #

# :. tan^2x -= sec^2x - 1#

Confirming that the result is an identity.

Jan 9, 2018

Yes, #sec^2-1=tan^2x# is an identity.

Explanation:

We start from #sin^2x+cos^2x=1#.

Then divide everything by #cos^2x#

#sin^2x/cos^2x+cos^2x/cos^2x=tan^2x+1=1/cos^2x=sec^2x#

Rearrange to find #tan^2x#

#tan^2x=sec^2x-1#