Question

Simplify tan(x)^2-tan(x)^2sin(x)^2

Answer 1

Hi, the first step would be to factor out what the terms have in common.

Answer 2

Do you see what they have in common?

Answer 3

tan(x)^2 and - tan(x)^2sin(x)^2 both have tan(x)^2 in common

Answer 4

tan(x)^2(1 - sin(x)^2)

Answer 5

\(\bf sin^2(\theta)+cos^2(\theta)=1\implies
cos^2(\theta)={\color{blue}{ 1-sin^2(\theta)}}\qquad tan(\theta)=\cfrac{sin(\theta)}{cos(\theta)}\\ \quad \\ \quad \\

tan(x)^2-tan(x)^2sin(x)^2\implies tan^2(x)[{\color{blue}{ 1-sin^2(x}})]\\ \quad \\
\cfrac{sin(\theta)}{cos(\theta)}[{\color{blue}{ 1-sin^2(x}})]\)

Answer 6

The next step is to solve the identity sin(x)^2 + cos(x)^2 = 1 for 1 - sin(x)^2

Answer 7

We can do this by subtracting sin(x)^2 from both sides.

Answer 8

hmmm actually

Answer 9

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

Answer 10

\(\bf sin^2(\theta)+cos^2(\theta)=1\implies
cos^2(\theta)={\color{blue}{ 1-sin^2(\theta)}}\qquad tan(\theta)=\cfrac{sin(\theta)}{cos(\theta)}\\ \quad \\ \quad \\

tan(x)^2-tan(x)^2sin(x)^2\implies tan^2(x)[{\color{blue}{ 1-sin^2(x}})]\\ \quad \\
\cfrac{sin^2(\theta)}{cos^2(\theta)}[{\color{blue}{ 1-sin^2(x}})]\) better

Answer 11

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

Answer 12

We can then substitute the 1 - sin(x)^2 with cos(x)^2

Answer 13

tan(x)^2(1 - sin(x)^2) = tan(x)^2(cos(x)^2)

Answer 14

Since tan(x) = sin(x)/cos(x) and tan(x)^2 = sin(x)^2/cos(x)^2, we can substitute again.

Answer 15

tan(x)^2(cos(x)^2) = (sin(x)^2/cos(x)^2)(cos(x)^2)

Answer 16

The cos(x)^2 cancels in the numerator and the denominator.

Answer 17

We are left with sin(x)^2

Answer 18

Thank you I understand what you are saying