How do you simplify tan2 x-1 over 1-sec2 x?

Answer 1

tan2 x + 1 = sec2 x

⇒ 1 - sec2 x = -tan2 x

⇒ (tan2 x - 1) / (1 - sec2 x) = (tan2 x - 1) / -tan2 x

= (tan2 x) / (-tan2 x) - 1 / (-tan2 x)

= -1 + cot2 x

= cot2 x - 1

If you cannot remember tan2 x + 1 = sec2 x, remember and start from:

sin2 x + cos2 x = 1

(which is used often) and divide each side by cos2 x:

(sin2 x + cos2 x) / cos2 x = 1 / cos2 x

⇒ sin2 x / cos2 x + cos2 x / cos2 x = 1 / cos2 x

But sin x / cos x = tan x; and 1/cos x = sec x

⇒ tan2 x + 1 = sec2 x

Also, cot x = 1/tan x