Question

Question 8 (5 points)

Simplify (tanx- secx) (tanx+ secx).
1
-1
sec^2xtan^2x
0

Answer 1

-1.

Explanation:

(tanx- secx) (tanx+ secx).

= tan^2 x + tanx sec x- tanx sec x - sec^2x

= tan^2 x - sec^2 x.

But sec^2 x = 1 + tan^2 x

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

Answer 2

-1 (negative one)

Explanation:

We operate as with a product of binomials, and once expanded we combine like terms:

`(tan(x)-sec(x))*(tan(x)+sec(x))=\\=tan^2(x)+tan(x)*sec(x)-sec(x)*tan(x)-sec^2(x)=\\=tan^2(x)-sec^2(x)`

and now we write tan and sec in terms of their basic trig expressions (
`tan(x)=(sin(x))/(cos(x))`, and
`sec(x)=(1)/(cos(x))`):

`tan^2(x)-sec^2(x)=(sin^2(x))/(cos^2(x)) -(1)/(cos^2(x)) =\\=(sin^2(x)-1)/(cos^2(x))`

From the Pythagorean identity:
`sin^2(x)+cos^2(x)=1`, we see that
`sin^2(x)-1=-cos^2(x)`, so we replace this in the expression above, so we are able to cancel the factor
`cos^2(x)`:

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