Question

How do you simplify `tan^2 4b - sec^2 4b`?

Answer 1

Call x = 4b

`f(x) = (tan x - sec x)(tan x + sec x)`

`= (sin x/cos x - 1/cos x)(sin x/cos x + 1/cos x)`

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

`= - (cos^2 x)/(cos^2 x) = -1`

Answer 2

An alternate solution using the Pythagorean Identity:

We know that `tan^2(x)+1=sec^2(x)`

Let's let `4b=x`

`tan^2(4b)+1=sec^2(4b)`

`tan^2(4b)-sec^2(4b) = -1`