Question

Question 8 (5 points) Simplify (tanx- secx) (tanx+ secx). 1 -1 sec^2xtan^2x 0

Answer 1

Answer:

-1 (negative one)

Step-by-step 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 ($tan(x)=\frac{sin(x)}{cos(x)}$, and $sec(x)=\frac{1}{cos(x)}$):

$tan^2(x)-sec^2(x)=\frac{sin^2(x)}{cos^2(x)} -\frac{1}{cos^2(x)} =\\=\frac{sin^2(x)-1}{cos^2(x)}$

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 $cos^2(x)$:

$\frac{sin^2(x)-1}{cos^2(x)}=\frac{-cos^2(x)}{cos^2(x)} =-1$

Answer 2

Answer:

-1.

Step-by-step 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