Answer:
[tex]sec^2\theta\cdot tan^2\theta[/tex]
Step-by-step explanation:
Trigonometric Functions
There are some basic relations between the trigonometric functions that allow us to transform and conveniently manage them if many different ways. Some basic identities are:
[tex]sin^2\theta+cos^2\theta=1[/tex]
[tex]sin(-\theta)=-sin\theta[/tex]
[tex]cos(-\theta)=cos\theta[/tex]
[tex]\displaystyle sec\theta=\frac{1}{cos\theta}[/tex]
[tex]sec(-\theta)=sec\theta[/tex]
We are required to simplify the following expression:
[tex]\displaystyle \frac{sec^2(-\theta)-1}{1-sin^2(-\theta)}[/tex]
Transforming the negative arguments
[tex]\displaystyle \frac{sec^2\theta-1}{1-sin^2\theta}[/tex]
Transforming the secant into cosine
[tex]\displaystyle \frac{\frac{1}{cos^2\theta}-1}{1-sin^2\theta}[/tex]
Operating
[tex]\displaystyle \frac{1}{cos^2\theta}\cdot \frac{1-cos^2\theta}{1-sin^2\theta}[/tex]
Applying the basic identity in the numerator and denominator
[tex]\displaystyle \frac{1}{cos^2\theta}\cdot \frac{sin^2\theta}{cos^2\theta}[/tex]
Since
[tex]\displaystyle tan\theta=\frac{sin\theta}{cos\theta}[/tex]
[tex]\boxed{sec^2\theta\cdot tan^2\theta}[/tex]
