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

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Answer 1 · 2015-06-02T02:36:40

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 · 2015-06-02T04:00:37

An alternate solution using the Pythagorean Identity:

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

Let's let $4b=x$

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

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

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